Home Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
Article Open Access

Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods

  • Keyan Wang EMAIL logo
Published/Copyright: August 27, 2024
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size H and the fine grid size h satisfy h = O ( H ( 2 k + 1 ) ⁄ ( k + 1 ) ) ( k ≥ 1 ), where k is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.

Keywords: hyperbolic equation; expanded mixed finite element method; two-grid method; error estimate
MSC 2010: 35L20; 65M60

1 Introduction

The mathematical model of certain physical phenomena, such as mathematical models of structural vibration, water waves, sound waves, and electromagnetic waves could be described as hyperbolic equations. Lots of numerical methods have been developed for solving these model problems [1–8]. In this article, we consider the following second-order hyperbolic equation:

(1.1) u t t − ∇ ⋅ ( K ∇ u ) = f ( u , ∇ u ) , ( x , t ) ∈ Ω × J , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × J ,

where Ω is a bounded polygonal domain in R 2 with boundary ∂ Ω , J = ( 0 , T ] and T > 0 is some final time. Let u t t and u t denote ∂ 2 u ∂ t 2 and ∂ u ∂ t , respectively. We assume that K is a square-integrable, symmetric, uniformly positive-definite tensor, and there exist constants K * , K * > 0 , for every x ∈ Ω ¯ , and any vector y ∈ R 2 , such that

(1.2) K * ∣ y ∣ 2 ≤ y T K ( x ) y ≤ K * ∣ y ∣ 2 .

We also assume that each element of f = f ( u , ∇ u ) is twice continuously differentiable with derivatives up to second order bounded earlier by F .

The expanded mixed finite element method (EMFEM) is a discretization technique for the partial differential equations. Indeed, the expanded mixed method expands the standard mixed formulation in the sense that three variables are explicitly treated, i.e., the unknown scalar, the gradient, and the flux. As a result, it is suitable for the case where the coefficient of differential equations is a small tensor. In the past few decades, this method has been developed and achieved some extensions. Chen [9,10] analyzed the linear/quasi-linear elliptic equations by EMFEM. Arbogast et al. [11] developed expanded mixed method based on the lowest order Raviart-Thomas-Nedelec element for linear elliptic problem. Woodward and Dawson [12] analyzed EMFEM approximations of nonlinear parabolic equation arising in flow in porous media. Gatica and Heuer [13] studied a linear second-order elliptic equation based on expanded mixed element variational formulation. Zhao et al. [7] proposed an H 1 -Galerkin mixed finite element procedure to deal with a nonlinear hyperbolic equation by combining the H 1 -Galerkin formulation and the EMFEM. Recently, Liu et al. [14] discussed a new expanded mixed method for parabolic integro-differential equations. Sharma et al. [15] applied the EMFEM for a class of nonlinear and nonlocal parabolic problems for the case of the lowest order RT element. In this article, an expanded mixed method for (1.1) is proposed and analyzed.

We introduce two variables: p ˜ = ∇ u and p = − K p ˜ , and then (1.1) can be transformed into the following formulation:

(1.3) u t t + ∇ ⋅ p = f ( u , p ˜ ) , ( x , t ) ∈ Ω × J , p ˜ − ∇ u = 0 , ( x , t ) ∈ Ω × J , p + K p ˜ = 0 , ( x , t ) ∈ Ω × J , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × J .

As we know, the resulting algebraic equations are large systems of nonlinear equations with the EMFEM for (1.1). So it is necessary for us to study a highly efficient and accurate algorithm for nonlinear system (1.3). We shall consider a two-grid method inspired by Xu [16,17]. The idea of two-grid method is using a coarse grid space to obtain a rough approximation of the solution for nonlinear problem and then correct it by solving a linear system on the fine grid space. Xu [16,17] mainly considered two-grid finite element method for nonsymmetric linear and nonlinear elliptic problems. After his work, two-grid method combined with other numerical methods was further investigated by many authors (see, e.g., [8,18–29]). Dawson and Wheeler [18] applied two-grid method combined with MFEM to nonlinear parabolic equation. Wu and Allen [19] presented a two-step two-grid algorithm for EMFEM of semilinear reaction-diffusion equations. Chen et al. [20–22] gave some multi-step two-grid algorithms using EMFEM for semilinear and nonlinear parabolic equations. Hu et al. [23,24] studied the error estimates of optimal order for the nonlinear miscible displacement problem with two-grid MFEM and two-grid EMFEM, respectively. Liu et al. [25] studied two-grid EMFEM for fourth-order reaction-diffusion problem with time-fractional derivative. They also investigated two-grid finite element method for nonlinear time-fractional cable equation in [26]. Chen and Liu [8] considered two-grid finite volume element method for nonlinear hyperbolic equation. However, as far as we know, there is no error analysis of two-grid EMFEM for nonlinear hyperbolic equation in the literature.

In this article, we propose a two-grid algorithm for solving the nonlinear problem (1.1) using EMFEM. Based on two Raviart-Thomas mixed element spaces, one coarse grid T H with mesh size H and one fine grid T h with mesh size h , solving a large nonlinear system of equations on the fine space is reduced to solving a linear problem on the fine space and a small nonlinear problem on the coarse space. It is showed that coarse space can be extremely coarse and we can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = O ( H ( 2 k + 1 ) ⁄ ( k + 1 ) ) .

The rest of this article is organized as follows. In Section 2, we present a two-grid algorithm combined with the fully discrete EMFEM for (1.1). In Section 3, we carry out the stability analysis for two-grid method. In Section 4, we deduce the error estimates for both the coarse grid and fine grid. In Section 5, we present the numerical results obtained by both the two-grid method and the EMFEM. Finally, we show some conclusions in the last section.

Throughout this article, let C denote a generic positive constant, which does not depend on the spatial mesh and temporal discretization parameters. Furthermore, we denote the standard Lebesgue space defined on Ω by L p ( Ω ) for p ≥ 1 , with norm ‖ ⋅ ‖ p . Let W m , p ( Ω ) be the standard Sobolev space with norm ‖ ⋅ ‖ m , p given by ‖ ϕ ‖ m , p p = ∑ ∣ κ ∣ ≤ m ‖ D κ ϕ ‖ L p ( Ω ) p . For p = 2 , we define H m ( Ω ) = W m , 2 ( Ω ) with norm ‖ ⋅ ‖ m = ‖ ⋅ ‖ m , 2 (the notation ‖ ⋅ ‖ = ‖ ⋅ ‖ 0 , 2 ).

2 Two-grid algorithm based on EMFEM

Let W = L 2 ( Ω ) and V = H ( div ; Ω ) . The expanded mixed weak formulation of (1.3) is to find ( u , p ˜ , p ) : J ↦ W × V × V such that

(2.1) ( u t t , w ) + ( ∇ ⋅ p , w ) = ( f ( u , p ˜ ) , w ) , ∀ w ∈ W ,

(2.2) ( p ˜ , v ) − ( ∇ u , v ) = 0 , ∀ v ∈ V ,

(2.3) ( p , v ) + ( K p ˜ , v ) = 0 , ∀ v ∈ V ,

with u ( x , 0 ) = u 0 ( x ) and u t ( x , 0 ) = u 1 ( x ) .

If we integrate by parts in (2.2) we obtain, for any v ∈ V ,

( ∇ u , v ) = ⟨ v ⋅ ν , u ⟩ − ( ∇ ⋅ v , u ) = − ( ∇ ⋅ v , u ) .

We consider finite-dimensional subspaces W h and V h of W and V (they may be Raviart-Thomas spaces of index k [ RT k ] [30] or Brezzi-Douglas-Marini spaces of index k [ BDM k ] [31]) associated with a quasi-uniform partition T h of Ω by triangles with diameter h . The following inclusion holds for the RT k spaces or BDM k spaces

(2.4) ∇ ⋅ v h ∈ W h , ∀ v h ∈ V h .

Let N be a positive integer, and let { t n : t n = n τ ; 0 ≤ n ≤ N } be a uniform partition of the time interval with the time step τ = T ⁄ N . For any function ϑ of time, let ϑ n denote ϑ ( t n ) . Set

ϑ n + 1 2 = 1 2 ( ϑ n + 1 + ϑ n ) , ∂ t ϑ n + 1 2 = ϑ n + 1 − ϑ n τ , ∂ t ϑ n = ϑ n + 1 − ϑ n − 1 2 τ , ∂ t t ϑ n = ϑ n + 1 − 2 ϑ n + ϑ n − 1 τ 2 .

Then, it is easy to verify the following relations:

∂ t ϑ n = ∂ t ϑ n + 1 2 + ∂ t ϑ n − 1 2 2 and ∂ t t ϑ n = ∂ t ϑ n + 1 2 − ∂ t ϑ n − 1 2 τ .

The fully discrete EMFEM for problem (2.1)–(2.3) is as follows: find ( u h n + 1 , p ˜ h n + 1 , p h n + 1 ) ∈ W h × V h × V h for 1 ≤ n ≤ N − 1 , such that

(2.5) ( ∂ t t u h n , w h ) + ( ∇ ⋅ p h n + 1 , w h ) = ( f ( u h n + 1 , p ˜ h n + 1 ) , w h ) , ∀ w h ∈ W h ,

(2.6) ( p ˜ h n + 1 , v h ) + ( ∇ ⋅ v h , u h n + 1 ) = 0 , ∀ v h ∈ V h ,

(2.7) ( K p ˜ h n + 1 , v h ) + ( p h n + 1 , v h ) = 0 , ∀ v h ∈ V h ,

with u h 0 = Q h u 0 ( x ) , u h 1 = Q h ( u 0 ( x ) + u 1 ( x ) τ + 1 2 u t t ( x , 0 ) τ 2 ) , where Q h can be the projection defined in (4.1).

In order to derive the two-grid expanded mixed finite element algorithm for the nonlinear hyperbolic equation (1.1), we introduce two quasi-uniform triangulations of Ω , T H and T h , with two different mesh sizes H and h ( h ≪ H < 1 ). The basic ingredient in our approach is another mixed finite element space W H × V H × V H ( ⊂ W h × V h × V h ) defined on a coarser mesh. The algorithm has the following two steps:

Step 1: On the coarse grid T H , find ( u H n + 1 , p ˜ H n + 1 , p H n + 1 ) ∈ W H × V H × V H ( 1 ≤ n ≤ N − 1 ) such that

(2.8) ( ∂ t t u H n , w H ) + ( ∇ ⋅ p H n + 1 , w H ) = ( f ( u H n + 1 , p ˜ H n + 1 ) , w H ) , ∀ w H ∈ W H ,

(2.9) ( p ˜ H n + 1 , v H ) + ( ∇ ⋅ v H , u H n + 1 ) = 0 , ∀ v H ∈ V H ,

(2.10) ( K p ˜ H n + 1 , v H ) + ( v H , p H n + 1 ) = 0 , ∀ v H ∈ V H ,

with u H 0 = Q H u 0 ( x ) , u H 1 = Q H ( u 0 ( x ) + u 1 ( x ) τ + 1 2 u t t ( x , 0 ) τ 2 ) .

Step 2: On the fine grid T h , find ( U h n + 1 , P ˜ h n + 1 , P h n + 1 ) ∈ W h × V h × V h ( 1 ≤ n ≤ N − 1 ) such that

(2.11) ( ∂ t t U h n , w h ) + ( ∇ ⋅ P h n + 1 , w h ) = ( f ( u H n + 1 , p ˜ H n + 1 ) + f u ( u H n + 1 , p ˜ H n + 1 ) ( U h n + 1 − u H n + 1 ) + f p ˜ ( u H n + 1 , p ˜ H n + 1 ) ⋅ ( P ˜ h n + 1 − p ˜ H n + 1 ) , w h ) , ∀ w h ∈ W h ,

(2.12) ( P ˜ h n + 1 , v h ) + ( ∇ ⋅ v h , U h n + 1 ) = 0 , ∀ v h ∈ V h ,

(2.13) ( K P ˜ h n + 1 , v h ) + ( v h , P h n + 1 ) = 0 , ∀ v h ∈ V h ,

with U h 0 = Q h u 0 ( x ) , U h 1 = Q h ( u 0 ( x ) + u 1 ( x ) τ + 1 2 u t t ( x , 0 ) τ 2 ) .

Remark 2.1

The solution ( u H n + 1 , p ˜ H n + 1 , p H n + 1 ) on the coarse grid T H for the nonlinear system (2.8)–(2.10) can be obtained through nonlinear iterations. Once ( u H n + 1 , p ˜ H n + 1 , p H n + 1 ) is known on the coarse grid T H , we can solve the linear system (2.11)–(2.13) on the fine grid T h to obtain ( U h n + 1 , P ˜ h n + 1 , P h n + 1 ). However, solving ( u h n + 1 , p ˜ h n + 1 , p h n + 1 ) directly from the nonlinear EMFE system (2.5)–(2.7) requires a significantly larger amount of computational time, as demonstrated by our numerical results in Section 5.

3 Stability analysis

In this section, we will give the stability analysis for two-grid scheme (2.8)–(2.13).

3.1 Stability analysis for the coarse grid

In order to derive the stability for our two-grid method, we need to obtain a stability result first for the coarse grid system (2.8)–(2.10).

Theorem 3.1

The scheme defined by (2.8)–(2.10) is stable for τ < 1 , and

( 2 − 2 τ ) ∂ t u H N − 1 2 2 + ‖ u H N ‖ 2 + 1 2 K 1 2 p ˜ H N 2 + θ 2 K * 2 ‖ p H N ‖ 2 ≤ 2 ∂ t u H 1 2 2 + K 1 2 p ˜ H 1 2 + K 1 2 p ˜ H 0 2 + C ‖ f n ‖ 2

holds, where 0 < θ ≤ K * .

Proof

Subtracting (2.9) from itself, with n + 1 replaced by n − 1 , we can easily obtain

(3.1) ( p ˜ H n + 1 − p ˜ H n − 1 , v H ) + ( u H n + 1 − u H n − 1 , ∇ ⋅ v H ) = 0 , ∀ v H ∈ V H .

Take w H = ∂ t u H n in (2.8), v H = ∂ t p ˜ H n in (2.10) and v H = p H n + 1 2 τ in (3.1), respectively, then combine the three resulting equations to obtain

(3.2) ( ∂ t t u H n , ∂ t u H n ) + ( K p ˜ H n + 1 , ∂ t p ˜ H n ) = ( f ( u H n + 1 , p ˜ H n + 1 ) , ∂ t u H n ) .

Note that

(3.3) − K 1 2 p ˜ H n + 1 2 + K 1 2 p ˜ H n − 1 2 ⁄ 2 ≤ − ( K p ˜ H n + 1 , p ˜ H n − 1 ) .

From (3.2) and (3.3), it easily follows that

(3.4) ( ∂ t t u H n , ∂ t u H n ) + ( K p ˜ H n + 1 , ∂ t p ˜ H n ) = 1 2 τ ( ∂ t u H n + 1 2 − ∂ t u H n − 1 2 , ∂ t u H n + 1 2 + ∂ t u H n − 1 2 ) + 1 2 τ ( K p ˜ H n + 1 , p ˜ H n + 1 − p ˜ H n − 1 ) ≥ 1 2 τ ∂ t u H n + 1 2 2 − ∂ t u H n − 1 2 2 + 1 4 τ K 1 2 p ˜ H n + 1 2 − K 1 2 p ˜ H n − 1 2 .

Applying the Cauchy-Schwarz inequality and Young’s inequality, we have

(3.5) ( f ( u H n + 1 , p ˜ H n + 1 ) , ∂ t u H n ) ≤ C ‖ f ( u H n + 1 , p ˜ H n + 1 ) ‖ ⋅ ‖ ∂ t u H n ‖ ≤ C ‖ f ( u H n + 1 , p ˜ H n + 1 ) ‖ 2 + 1 2 ‖ ∂ t u H n ‖ 2 .

Substituting (3.4) and (3.5) into (3.2), we see that

(3.6) 1 2 τ ( ∂ t u H n + 1 2 2 − ∂ t u H n − 1 2 2 ) + 1 4 τ ( K 1 2 p ˜ H n + 1 2 − K 1 2 p ˜ H n − 1 2 ) ≤ C ‖ f ( u H n + 1 , p ˜ H n + 1 ) ‖ 2 + 1 2 ‖ ∂ t u H n ‖ 2 .

Multiplying both sides of (3.6) by 4 τ and summing from 1 to N − 1 , we obtain

(3.7) 2 ∂ t u H N − 1 2 2 + K 1 2 p ˜ H N 2 + K 1 2 p ˜ H N − 1 2 ≤ 2 ∂ t u H 1 2 2 + K 1 2 p ˜ H 1 2 + K 1 2 p ˜ H 0 2 + 2 τ ∑ n = 1 N − 1 ‖ ∂ t u H n ‖ 2 + C τ ∑ n = 1 N − 1 ‖ f ( u H n + 1 , p ˜ H n + 1 ) ‖ 2 ≤ 2 ∂ t u H 1 2 2 + K 1 2 p ˜ H 1 2 + K 1 2 p ˜ H 0 2 + 2 τ ∑ n = 1 N − 1 ‖ ∂ t u H n ‖ 2 + C ‖ f n ‖ 2 ,

where ‖ f n ‖ = max { ‖ f ( u H 1 , p ˜ H 1 ) ‖ , ‖ f ( u H 2 , p ˜ H 2 ) ‖ , … , ‖ f ( u H N , p ˜ H N ) ‖ } . Let v H = p H n + 1 ∈ V H in (2.10), and we obtain

‖ p H n + 1 ‖ ≤ K * ‖ p ˜ H n + 1 ‖ .

Therefore,

(3.8) θ 2 K * 2 ‖ p H n + 1 ‖ 2 ≤ θ 2 ‖ p ˜ H n + 1 ‖ 2 ≤ 1 2 K 1 2 p ˜ H n + 1 2 ,

where 0 < θ ≤ K * . In view of

(3.9) ‖ ∂ t u H n ‖ 2 ≤ ∂ t u H n + 1 2 2 + ∂ t u H n − 1 2 2 .

Combining the estimates (3.7)–(3.9), we have

(3.10) ( 2 − 2 τ ) ∂ t u H N − 1 2 2 + 1 2 K 1 2 p ˜ H N 2 + K 1 2 p ˜ H N − 1 2 + θ 2 K * 2 ‖ p H N ‖ 2 ≤ 2 ∂ t u H 1 2 2 + K 1 2 p ˜ H 1 2 + K 1 2 p ˜ H 0 2 + 2 τ ∑ n = 2 N − 1 ∂ t u H n − 1 2 2 + C ‖ f n ‖ 2 .

Adding a term ‖ u H N ‖ 2 at the two sides of (3.10) and using the inequality,

‖ u H N ‖ 2 ≤ C τ ∑ n = 1 N u H n − 1 2 2 .

When τ < 1 , the discrete Gronwall’s lemma gives

(3.11) ( 2 − 2 τ ) ∂ t u H N − 1 2 2 + ‖ u H N ‖ 2 + 1 2 K 1 2 p ˜ H N 2 + θ 2 K * 2 ‖ p H N ‖ 2 ≤ 2 ∂ t u H 1 2 2 + K 1 2 p ˜ H 1 2 + K 1 2 p ˜ H 0 2 + C ‖ f n ‖ 2 .

Hence, we complete the proof of Theorem 3.1.□

3.2 Stability analysis for the fine grid

The following stability result can be proved by the same argument as used for Theorem 3.1.

Theorem 3.2

For 0 < θ ≤ K * , if τ < min 1 3 + F 2 + F 2 θ , 1 4 , where F > 0 represents an upper bound for derivatives of function f, then for the fine grid scheme (2.11)–(2.13), the following stability holds true:

(3.12) 2 − 2 τ 3 + F 2 + F 2 θ ∂ t U h N − 1 2 2 + ( 1 − 2 τ ) ‖ U h N ‖ 2 + θ 2 K * 2 ‖ P h N ‖ 2 + 1 2 − 2 τ K 1 2 P ˜ h N 2 ≤ 2 ∂ t U h 1 2 2 + K 1 2 P ˜ h 1 2 + K 1 2 P ˜ h 0 2 + C ( ∂ t u H 1 2 2 + K 1 2 p ˜ H 1 2 + K 1 2 p ˜ H 0 2 + ‖ f n ‖ 2 ) .

Proof

Similarly, as in Theorem 3.1, use the boundedness assumption on f , and we have (cf. (3.6)):

1 2 τ ∂ t U h n + 1 2 2 − ∂ t U h n − 1 2 2 + 1 4 τ K 1 2 P ˜ h n + 1 2 − K 1 2 P ˜ h n − 1 2 ≤ ( f ( u H n + 1 , p ˜ H n + 1 ) + f u ( u H n + 1 , p ˜ H n + 1 ) ( U h n + 1 − u H n + 1 ) + f p ˜ ( u H n + 1 , p ˜ H n + 1 ) ⋅ ( P ˜ h n + 1 − p ˜ H n + 1 ) , ∂ t U h n ) ≤ C ‖ u H n + 1 ‖ 2 + K 1 2 p ˜ H n + 1 2 + 1 2 ‖ U h n + 1 ‖ 2 + 1 2 K 1 2 P ˜ h n + 1 2 + 3 + F 2 2 + F 2 2 θ ‖ ∂ t U h n ‖ 2 + C ‖ f ( u H n + 1 , p ˜ H n + 1 ) ‖ 2 .

Following a similar analysis as that carried out for (3.10), we can obtain

2 ∂ t U h N − 1 2 2 + ‖ U h N ‖ 2 + 1 2 K 1 2 P ˜ h N 2 + K 1 2 P ˜ h N − 1 2 + θ 2 K * 2 ‖ P h N ‖ 2 ≤ 2 ∂ t U h 1 2 2 + K 1 2 P ˜ h 1 2 + K 1 2 P ˜ h 0 2 + 2 τ ∑ n = 1 N − 1 ‖ U h n + 1 ‖ 2 + 2 τ ∑ n = 1 N − 1 K 1 2 P ˜ h n + 1 2 + 2 τ 3 + F 2 + F 2 θ ∑ n = 1 N − 1 ∂ t U h n + 1 2 2 + C τ ∑ n = 1 N U h n − 1 2 2 + C ‖ f n ‖ 2 + C τ ∑ n = 1 N − 1 ( ‖ u H n + 1 ‖ 2 + K 1 2 p ˜ H n + 1 2 ) .

Use (3.11) and Gronwall’s lemma to obtain

2 − 2 τ 3 + F 2 + F 2 θ ∂ t U h N − 1 2 2 + ( 1 − 2 τ ) ‖ U h N ‖ 2 + 1 2 − 2 τ K 1 2 P ˜ h N 2 + K 1 2 P ˜ h N − 1 2 + θ 2 K * 2 ‖ P h N ‖ 2 ≤ 2 ∂ t U h 1 2 2 + K 1 2 P ˜ h 1 2 + K 1 2 P ˜ h 0 2 + C τ ∑ n = 1 N − 1 ( ∂ t u H 1 2 2 + K 1 2 p ˜ H 1 2 + K 1 2 p ˜ H 0 2 ) + C ‖ f n ‖ 2 .

Note that τ ∑ n = 1 N − 1 ≤ T . The desired inequality (3.12) follows from the aforementioned inequality, and the proof is complete.□

4 Error analysis based on two-grid algorithm

For discussing and deriving a priori error estimate based on fully discrete two-grid EMFEM, we employ some L 2 projection operators such as Q ℏ : W → W ℏ and R ℏ : V → V ℏ , satisfying

(4.1) ( u , w ℏ ) = ( Q ℏ u , w ℏ ) , ∀ w ℏ ∈ W ℏ ,

(4.2) ( p ˜ , v ℏ ) = ( R ℏ p ˜ , v ℏ ) , ∀ v ℏ ∈ V ℏ ,

where ℏ is either h or H depending on whether we work on the fine grid space or coarse grid space.

Next, recall the Fortin projection Π ℏ : ( H 1 ( Ω ) ) 2 → V ℏ , which satisfies: for any p ∈ V ,

(4.3) ( ∇ ⋅ p , w ℏ ) = ( ∇ ⋅ Π ℏ p , w ℏ ) , ∀ w ℏ ∈ W ℏ .

The following approximation properties [30,31] hold for projections Q ℏ , R ℏ , and Π ℏ :

(4.4) ‖ Q ℏ u ‖ 0 , q ≤ C ‖ u ‖ 0 , q , 2 ≤ q < ∞ ,

(4.5) ‖ R ℏ p ˜ ‖ 0 , q ≤ C ‖ p ˜ ‖ 0 , q , 2 ≤ q < ∞ ,

(4.6) ‖ u − Q ℏ u ‖ 0 , q ≤ C ‖ u ‖ r , q ℏ r , 0 ≤ r ≤ k + 1 ,

(4.7) ‖ p ˜ − R ℏ p ˜ ‖ 0 , q ≤ C ‖ p ˜ ‖ r , q ℏ r , 0 ≤ r ≤ k + 1 ,

(4.8) ‖ p − Π ℏ p ‖ 0 , q ≤ C ‖ p ‖ r , q ℏ r , 1 ⁄ q < r ≤ k + 1 ,

(4.9) ‖ ∇ ⋅ ( p − Π ℏ p ) ‖ 0 , q ≤ C ‖ ∇ ⋅ p ‖ r , q ℏ r , 0 ≤ r ≤ k + 1 .

4.1 Error analysis for the coarse grid

To simplify the error analysis process, we split the errors into the following:

u n − u H n = u n − Q H u n + Q H u n − u H n = ξ n + μ n , p ˜ n − p ˜ H n = p ˜ n − R H p ˜ n + R H p ˜ n − p ˜ H n = η ¯ n + χ ¯ n , p n − p H n = p n − Π H p n + Π H p n − p H n = η n + χ n .

In the following, we will present a detailed proof of a priori error estimate for the coarse grid system (2.8)–(2.10).

Theorem 4.1

Let ( u n , p ˜ n , p n ) and ( u H n , p ˜ H n , p H n ) be the solutions of (2.1)–(2.3) and (2.8)–(2.10), respectively. Take 0 < θ ≤ K * . If τ < min 1 2 ( 3 + F 2 2 + F 2 θ ) , 1 4 , where F > 0 represents an upper bound for derivatives of function f, then for 1 ≤ n ≤ N , we have

(4.10) sup n { ∣ ∣ u n − u H n ∣ ∣ + ∣ ∣ p ˜ n − p ˜ H n ∣ ∣ + ∣ ∣ p n − p H n ∣ ∣ } ≤ C ( H k + 1 + τ 2 ) ,

where k is associated with the degree of the finite element polynomial.

Proof

At the time of t = t n + 1 , using the projection identities of Q H , R H , and Π H , and assumption (2.4), we rewrite (2.1)–(2.3) in the following form:

(4.11) ( ∂ t t Q H u n , w H ) + ( ∇ ⋅ Π H p n + 1 , w H ) = ( f ( u n + 1 , p ˜ n + 1 ) , w H ) + ( ε n + 1 , w H ) , ∀ w H ∈ W H ,

(4.12) ( R H p ˜ n + 1 , v H ) + ( ∇ ⋅ v H , Q H u n + 1 ) = 0 , ∀ v H ∈ V H ,

(4.13) ( Π H p n + 1 , v H ) = ( Π H p n + 1 − p n + 1 , ∀ v H ) − ( K p ˜ n + 1 , v H ) , ∀ v H ∈ V H ,

where ε n + 1 is a time truncation error of order τ 2 .

Subtracting (2.5)–(2.7) from (4.11)–(4.13), we have the following error equations:

(4.14) ( ∂ t t μ n , w H ) + ( ∇ ⋅ χ n + 1 , w H ) = ( f ( u n + 1 , p ˜ n + 1 ) − f ( u H n + 1 , p ˜ H n + 1 ) , w H ) + ( ε n + 1 , w H ) , ∀ w H ∈ W H ,

(4.15) ( χ ¯ n + 1 , v H ) + ( μ n + 1 , ∇ ⋅ v H ) = 0 , ∀ v H ∈ V H ,

(4.16) ( K χ ¯ n + 1 , v H ) + ( χ n + 1 , v H ) = − ( η n + 1 , v H ) − ( K η ¯ n + 1 , v H ) , ∀ v H ∈ V H .

Take the difference in time of (4.15) to obtain

(4.17) ( ∂ t χ ¯ n , v H ) + ( ∂ t μ n , ∇ ⋅ v H ) = 0 , ∀ v H ∈ V H .

Choose w H = ∂ t μ n , v H = ∂ t χ ¯ n , and v H = χ n + 1 in (4.14), (4.16), and (4.17), respectively. We add (4.14) and (4.16), and then, subtract (4.17) to reduce to a single equation:

(4.18) ( ∂ t t μ n , ∂ t μ n ) + ( K χ ¯ n + 1 , ∂ t χ ¯ n ) = ( ε n + 1 , ∂ t μ n ) + ( f ( u n + 1 , p ˜ n + 1 ) − f ( u H n + 1 , p ˜ H n + 1 ) , ∂ t μ n ) − ( η n + 1 + K η ¯ n + 1 , ∂ t χ ¯ n ) .

Note that in (3.3), the terms in the left-hand side of (4.18) can be estimated as

(4.19) ( ∂ t t μ n , ∂ t μ n ) + ( K χ ¯ n + 1 , ∂ t χ ¯ n ) = 1 2 τ ∂ t μ n + 1 2 − ∂ t μ n − 1 2 , ∂ t μ n + 1 2 + ∂ t μ n − 1 2 + 1 2 τ ( K χ ¯ n + 1 , χ ¯ n + 1 − χ ¯ n − 1 ) ≥ 1 2 τ ∂ t μ n + 1 2 2 − ∂ t μ n − 1 2 2 + 1 4 τ K 1 2 χ ¯ n + 1 2 − K 1 2 χ ¯ n − 1 2 .

To consider the second term in the right-hand side of (4.18), we use Taylor expansion and the boundedness assumption on f to obtain

(4.20) ( f ( u n + 1 , p ˜ n + 1 ) − f ( u H n + 1 , p ˜ H n + 1 ) , ∂ t μ n ) ≤ ( F ξ n + 1 , ∂ t μ n ) + ( F μ n + 1 , ∂ t μ n ) + ( F η ¯ n + 1 , ∂ t μ n ) + ( F χ ¯ n + 1 , ∂ t μ n ) .

Using Cauchy’s inequality, we have

(4.21) ( ε n + 1 , ∂ t μ n ) + ( F ξ n + 1 , ∂ t μ n ) + ( F μ n + 1 , ∂ t μ n ) + ( F η ¯ n + 1 , ∂ t μ n ) + ( F χ ¯ n + 1 , ∂ t μ n ) ≤ 1 2 ‖ ε n + 1 ‖ 2 + 1 2 ‖ ∂ t μ n ‖ 2 + F 2 2 ‖ ξ n + 1 ‖ 2 + 1 2 ‖ ∂ t μ n ‖ 2 + 1 2 ‖ μ n + 1 ‖ 2 + F 2 2 ‖ ∂ t μ n ‖ 2 + F 2 2 ‖ η ¯ n + 1 ‖ 2 + 1 2 ‖ ∂ t μ n ‖ 2 + θ 2 ‖ χ ¯ n + 1 ‖ 2 + F 2 2 θ ‖ ∂ t μ n ‖ 2 ≤ 1 2 ‖ ε n + 1 ‖ 2 + 3 + F 2 2 + F 2 2 θ ‖ ∂ t μ n ‖ 2 + 1 2 ‖ μ n + 1 ‖ 2 + 1 2 K 1 2 χ ¯ n + 1 2 + C ( ‖ ξ n + 1 ‖ 2 + ‖ η ¯ n + 1 ‖ 2 ) ,

where 0 < θ ≤ K * . Using the Cauchy-Schwarz inequality and ε -Cauchy inequality, we obtain

(4.22) ( η n + 1 + K η ¯ n + 1 , ∂ t χ ¯ n ) ≤ C ( ‖ η n + 1 ‖ + ‖ η ¯ n + 1 ‖ ) ‖ ∂ t χ ¯ n ‖ ≤ C ( ‖ η n + 1 ‖ 2 + ‖ η ¯ n + 1 ‖ 2 ) + ε ‖ ∂ t χ ¯ n ‖ 2 ,

for any small constant ε > 0 . Combining (4.18)–(4.22), we find that

(4.23) 1 2 τ ( ∂ t μ n + 1 2 2 − ∂ t μ n − 1 2 2 ) + 1 4 τ ( K 1 2 χ ¯ n + 1 2 − K 1 2 χ ¯ n − 1 2 ) ≤ 1 2 ‖ ε n + 1 ‖ 2 + 3 + F 2 2 + F 2 2 θ ‖ ∂ t μ n ‖ 2 + 1 2 ‖ μ n + 1 ‖ 2 + 1 2 K 1 2 χ ¯ n + 1 2 + C ( ‖ ξ n + 1 ‖ 2 + ‖ η n + 1 ‖ 2 + ‖ η ¯ n + 1 ‖ 2 ) .

Multiplying by 4 τ and summing over n from 1 to N − 1 at both sides of (4.23), we see that

(4.24) 2 ∂ t μ N − 1 2 2 − 2 ∂ t μ 1 2 2 + K 1 2 χ ¯ N 2 + K 1 2 χ ¯ N − 1 2 − K 1 2 χ ¯ 1 2 − K 1 2 χ ¯ 0 2 ≤ 4 τ ∑ n = 1 N − 1 C ( ‖ ξ n + 1 ‖ 2 + ‖ η n + 1 ‖ 2 + ‖ η ¯ n + 1 ‖ 2 ) + 2 τ ∑ n = 1 N − 1 ‖ ε n + 1 ‖ 2 + 2 τ ∑ n = 1 N − 1 K 1 2 χ ¯ n + 1 2 + 2 τ ∑ n = 1 N − 1 ‖ μ n + 1 ‖ 2 + 4 τ 3 + F 2 2 + F 2 2 θ ∑ n = 1 N − 1 ‖ ∂ t μ n ‖ 2 .

Note that Q H u 0 = u H 0 , Q H u 1 − u H 1 = C τ 3 , and ‖ χ ¯ 1 ‖ 2 + ∂ t μ 1 2 2 ≤ C τ 4 ; thus by (3.9) and combining the error estimates (4.6)–(4.8) and (4.24) to obtain

(4.25) 2 − 4 τ 3 + F 2 2 + F 2 2 θ ∂ t μ N − 1 2 2 + ( 1 − 2 τ ) K 1 2 χ ¯ N 2 + K 1 2 χ ¯ N − 1 2 ≤ C ( H 2 k + 2 + τ 4 ) + 4 τ 3 + F 2 2 + F 2 2 θ ∑ n = 2 N − 1 ∂ t μ n − 1 2 2 + 2 τ ∑ n = 1 N − 1 ‖ μ n + 1 ‖ 2 .

Similar to (3.8), we have

(4.26) θ 2 K * 2 ‖ χ N ‖ 2 ≤ 1 2 K 1 2 χ ¯ N 2 .

Combining (4.25) with (4.26) leads to

(4.27) 2 − 4 τ 3 + F 2 2 + F 2 2 θ ∂ t μ N − 1 2 2 + 1 2 − 2 τ K 1 2 χ ¯ N 2 + θ 2 K * 2 ‖ χ N ‖ 2 ≤ C ( H 2 k + 2 + τ 4 ) + 4 τ 3 + F 2 2 + F 2 2 θ ∑ n = 2 N − 1 ∂ t μ n − 1 2 2 + 2 τ ∑ n = 1 N − 1 ‖ μ n + 1 ‖ 2 .

Adding a term ‖ μ N ‖ 2 at the two sides of (4.27) and using the inequality,

‖ μ N ‖ 2 ≤ C Δ t ∑ n = 1 N μ n − 1 2 2 .

When 2 − 4 τ 3 + F 2 2 + F 2 2 θ > 0 and 1 2 − 2 τ > 0 , i.e.,

τ < min 1 2 3 + F 2 2 + F 2 2 θ , 1 4 ,

with the application of discrete Gronwall’s lemma, we obtain the following result:

(4.28) ∂ t μ N − 1 2 2 + ‖ μ N ‖ 2 + ‖ χ N ‖ 2 + K 1 2 χ ¯ N 2 ≤ C ( H 2 k + 2 + τ 4 ) .

Then, by (4.6), (4.8), (4.28), and the triangle inequality, we can derive (4.10).□

4.2 Error analysis for the fine grid

In this subsection, we can prove the following theorem for the solution of the fine grid.

Theorem 4.2

Define the triplet ( U h n , P ˜ h n , P h n ) ∈ W h × V h × V h by (2.11)–(2.13). Take 0 < θ ≤ K * , when τ < min 1 2 3 + F 2 2 + F 2 2 θ , 1 4 , where F > 0 represents an upper bound for derivatives of function f, and for 1 ≤ n ≤ N and k ≥ 1 , we have the following error estimate:

sup n { ∣ ∣ u n − U h n ∣ ∣ + ∣ ∣ p ˜ n − P ˜ h n ∣ ∣ + ∣ ∣ p n − P h n ∣ ∣ } ≤ C ( h k + 1 + H 2 k + 1 + τ 2 ) ,

where k is associated with the degree of the finite element polynomial.

Proof

Let β n = Q h u n − U h n , γ ¯ n = R h p ˜ n − P ˜ h n , γ n = Π h p n − P h n , ξ * n = u n − Q h u n , η ¯ * n = p ˜ n − R h p ˜ n , and η * n = p n − Π h p n . From (2.1)–(2.3) and (2.11)–(2.13), and by the properties of the L 2 and Π ℏ projections, we can write

(4.29) ( ∂ t t β n , w h ) + ( ∇ ⋅ γ n + 1 , w h ) = ( ε n + 1 , w h ) + ( f ( u n + 1 , p ˜ n + 1 ) , w h ) − ( f ( u H n + 1 , p ˜ H n + 1 ) + f u ( u H n + 1 , p ˜ H n + 1 ) ( U h n + 1 − u H n + 1 ) , w h ) − ( f p ˜ ( u H n + 1 , p ˜ H n + 1 ) ⋅ ( P ˜ h n + 1 − p ˜ H n + 1 ) , w h ) , ∀ w h ∈ W h ,

(4.30) ( γ ¯ n + 1 , v h ) + ( β n + 1 , ∇ ⋅ v h ) = 0 , ∀ v h ∈ V h ,

(4.31) ( K γ ¯ n + 1 , v h ) + ( γ n + 1 , v h ) = − ( η * n + 1 , v h ) − ( K η ¯ * n + 1 , v h ) , ∀ v h ∈ V h .

We define the following notations: u s n + 1 = u H n + 1 + s ( u n + 1 − u H n + 1 ) , p ˜ s n + 1 = p ˜ H n + 1 + s ( p ˜ n + 1 − p ˜ H n + 1 ) , 0 ≤ s ≤ 1 , and employ the following Taylor expansion:

f ( u n + 1 , p ˜ n + 1 ) = f ( u H n + 1 , p ˜ H n + 1 ) + f u ( u H n + 1 , p ˜ H n + 1 ) ( u n + 1 − u H n + 1 ) + f p ˜ ( u H n + 1 , p ˜ H n + 1 ) ⋅ ( p ˜ n + 1 − p ˜ H n + 1 ) + f ˜ u u ( u n + 1 − u H n + 1 ) 2 + 2 f ˜ u p ˜ ⋅ ( u n + 1 − u H n + 1 ) ( p ˜ n + 1 − p ˜ H n + 1 ) + ( p ˜ n + 1 − p ˜ H n + 1 ) T f ˜ p ˜ p ˜ ( p ˜ n + 1 − p ˜ H n + 1 ) ,

where

f ˜ u u = ∫ 0 1 ( 1 − s ) f u u ( u s n + 1 , p ˜ s n + 1 ) d s , f ˜ u p ˜ = ∫ 0 1 ( 1 − s ) f u p ˜ ( u s n + 1 , p ˜ s n + 1 ) d s , f ˜ p ˜ p ˜ = ∫ 0 1 ( 1 − s ) f p ˜ p ˜ ( u s n + 1 , p ˜ s n + 1 ) d s .

Take the difference in time of (4.30) to obtain

(4.32) ( ∂ t γ ¯ n , v h ) + ( ∂ t β n , ∇ ⋅ v h ) = 0 , ∀ v h ∈ V h .

Choose w h = ∂ t β n , v h = ∂ t γ ¯ n , and v h = γ n + 1 in (4.29), (4.31), and (4.32), respectively. Then, adding (4.29) to (4.31), and subtracting (4.32), we can obtain

(4.33) 1 2 τ ∂ t β n + 1 2 2 − ∂ t β n − 1 2 2 + 1 4 τ K 1 2 γ ¯ n + 1 2 − K 1 2 γ ¯ n − 1 2 ≤ ( ∂ t t β n , ∂ t β n ) + ( K γ ¯ n + 1 , ∂ t γ ¯ n ) = ( f u ( u H n + 1 , p ˜ H n + 1 ) ( ξ * n + 1 + β n + 1 ) , ∂ t β n ) + ( f p ˜ ( u H n + 1 , p ˜ H n + 1 ) ⋅ ( η ¯ * n + 1 + γ ¯ n + 1 ) , ∂ t β n ) + ( f ˜ u u ( u n + 1 − u H n + 1 ) 2 , ∂ t β n ) + ( 2 f ˜ u p ˜ ⋅ ( u n + 1 − u H n + 1 ) ( p ˜ n + 1 − p ˜ H n + 1 ) , ∂ t β n ) + ( ( p ˜ n + 1 − p ˜ H n + 1 ) T f ˜ p ˜ p ˜ ( p ˜ n + 1 − p ˜ H n + 1 ) , ∂ t β n ) + ( ε n + 1 , ∂ t β n ) − ( η * n + 1 , ∂ t γ ¯ n ) − ( K η ¯ * n + 1 , ∂ t γ ¯ n ) = I 1 + I 2 .

Now, we estimate the right-hand terms of (4.33). For I 1 , applying the boundedness assumption on f and the Hölder’s inequality, we find that

I 1 = ( f u ( u H n + 1 , p ˜ H n + 1 ) ( ξ * n + 1 + β n + 1 ) , ∂ t β n ) + ( f p ˜ ( u H n + 1 , p ˜ H n + 1 ) ⋅ ( η ¯ * n + 1 + γ ¯ n + 1 ) , ∂ t β n ) + ( f ˜ u u ( u n + 1 − u H n + 1 ) 2 , ∂ t β n ) + ( 2 f ˜ u p ˜ ⋅ ( u n + 1 − u H n + 1 ) ( p ˜ n + 1 − p ˜ H n + 1 ) , ∂ t β n ) + ( ( p ˜ n + 1 − p ˜ H n + 1 ) T f ˜ p ˜ p ˜ ( p ˜ n + 1 − p ˜ H n + 1 ) , ∂ t β n ) + ( ε n + 1 , ∂ t β n ) ≤ C ( ‖ ξ * n + 1 ‖ 2 + ‖ η ¯ * n + 1 ‖ 2 + ‖ u n + 1 − u H n + 1 ‖ 0 , 4 4 + ‖ p ˜ n + 1 − p ˜ H n + 1 ‖ 0 , 4 4 ) + 1 2 ‖ β n + 1 ‖ 2 + θ 2 ‖ γ ¯ n + 1 ‖ 2 + 1 2 ‖ ε n + 1 ‖ 2 + 3 + F 2 2 + F 2 2 θ ‖ ∂ t β n ‖ 2 ,

where 0 < θ ≤ K * . For I 2 , by the ε -Cauchy inequality, we have

I 2 ≤ ∣ ( η * n + 1 , ∂ t γ ¯ n ) ∣ + ∣ ( K η ¯ * n + 1 , ∂ t γ ¯ n ) ∣ ≤ C ( ‖ η * n + 1 ‖ 2 + ‖ η ¯ * n + 1 ‖ 2 ) + ε ‖ ∂ t γ ¯ n ‖ 2 ,

for any small constant ε > 0 . Therefore, substituting these estimates into (4.33), we see that

(4.34) 1 2 τ ( ∂ t β n + 1 2 2 − ∂ t β n − 1 2 2 ) + 1 4 τ ( K 1 2 γ ¯ n + 1 2 − K 1 2 γ ¯ n − 1 2 ) ≤ C ( ‖ ξ * n + 1 ‖ 2 + ‖ η * n + 1 ‖ 2 + ‖ η ¯ * n + 1 ‖ 2 + ‖ u n + 1 − u H n + 1 ‖ 0 , 4 4 + ‖ p ˜ n + 1 − p ˜ H n + 1 ‖ 0 , 4 4 ) + 1 2 ‖ β n + 1 ‖ 2 + θ 2 ‖ γ ¯ n + 1 ‖ 2 + 1 2 ‖ ε n + 1 ‖ 2 + 3 + F 2 2 + F 2 2 θ ‖ ∂ t β n ‖ 2 .

Summing (4.34) for n from 1 to N − 1 and multiplying by 4 τ give

(4.35) 2 ∂ t β N − 1 2 2 − 2 ∂ t β 1 2 2 + K 1 2 γ ¯ N 2 + K 1 2 γ ¯ N − 1 2 − K 1 2 γ ¯ 1 2 − K 1 2 γ ¯ 0 2 ≤ C τ ∑ n = 1 N − 1 ( ‖ ξ * n + 1 ‖ 2 + ‖ η * n + 1 ‖ 2 + ‖ η ¯ * n + 1 ‖ 2 + ‖ u n + 1 − u H n + 1 ‖ 0 , 4 4 + ‖ p ˜ n + 1 − p ˜ H n + 1 ‖ 0 , 4 4 ) + 4 τ ∑ n = 1 N − 1 1 2 ‖ β n + 1 ‖ 2 + θ 2 ‖ γ ¯ n + 1 ‖ 2 + 1 2 ‖ ε n + 1 ‖ 2 + 3 + F 2 2 + F 2 2 θ ‖ ∂ t β n ‖ 2 .

In the following, similar to the estimate of (3.8), we have

(4.36) θ 2 K * 2 ‖ γ N ‖ 2 ≤ 1 2 K 1 2 γ ¯ N 2 .

With the help of (3.9), combine (4.35) with (4.36) and note that ‖ γ ¯ 1 ‖ 2 + ∂ t β 1 2 2 ≤ C τ 4 to obtain

2 − 4 τ 3 + F 2 2 + F 2 2 θ ∂ t β N − 1 2 2 + θ 2 K * 2 ‖ γ N ‖ 2 + 1 2 − 2 τ K 1 2 γ ¯ N 2 + K 1 2 γ ¯ N − 1 2 ≤ C τ ∑ n = 1 N − 1 ‖ ξ * n + 1 ‖ 2 + ‖ η * n + 1 ‖ 2 + ‖ η ¯ * n + 1 ‖ 2 + ‖ u n + 1 − u H n + 1 ‖ 0 , 4 4 + ‖ p ˜ n + 1 − p ˜ H n + 1 ‖ 0 , 4 4 + 2 τ ∑ n = 1 N − 1 ‖ β n + 1 ‖ 2 + 4 τ ∑ n = 2 N − 1 1 2 K 1 2 γ ¯ n 2 + 3 + F 2 2 + F 2 2 θ ∂ t β n − 1 2 2 + 2 τ ∑ n = 1 N − 1 ‖ ε n + 1 ‖ 2 ≤ 2 τ ∑ n = 1 N − 1 ‖ β n + 1 ‖ 2 + 4 τ ∑ n = 2 N − 1 1 2 K 1 2 γ ¯ n 2 + 3 + F 2 2 + F 2 2 θ ∂ t β n − 1 2 2 + C ( h 2 k + 2 + H 4 k + 2 + τ 4 ) ,

where we use (4.6)–(4.8), (4.10), and inverse assumption. Take

τ < min 1 2 3 + F 2 2 + F 2 2 θ , 1 4 ,

use the inequality ‖ β N ‖ 2 ≤ C Δ t ∑ n = 1 N ‖ β n − 1 2 ‖ 2 , and apply Gronwall’s lemma to obtain

(4.37) ∂ t β N − 1 2 2 + ‖ β N ‖ 2 + ‖ γ N ‖ 2 + K 1 2 γ ¯ N 2 ≤ C ( h 2 k + 2 + H 4 k + 2 + τ 4 ) .

Combining (4.37), the estimates of projections (4.6)–(4.8), and the triangle inequality leads to the final result.□

5 Numerical example

In this section, we give a numerical experiment to illustrate the efficiency and accuracy of the two-grid method. The following two-dimensional hyperbolic problem is considered:

(5.1) u t t − ∇ ⋅ ( K ( x ) ∇ u ) = f 1 ( u , ∇ u ) + f 2 ( x , t ) , ( x , t ) ∈ Ω × J , u ( x , 0 ) = u 0 , u t ( x , 0 ) = u 1 , x ∈ Ω , u = 0 , ( x , t ) ∈ ∂ Ω × J ,

where Ω ¯ = [ 0 , 1 ] 2 , J = ( 0 , 1 ] , x = ( x 1 , x 2 ) T , and K ( x ) = x 1 2 + 1 0 0 x 2 2 + 1 . The functions f 2 , u 0 and u 1 are computed from the exact solution u and the expression f 1 ( u , ∇ u ) . The exact solution u and f 1 ( u , ∇ u ) are chosen as follows:

f 1 ( u , ∇ u ) = ( b ⋅ ∇ u ) 2 + u 3 − u , b = ( 1 , 1 ) T , u = t 2 cos ( 2 π t ) sin ( π x 1 ) sin ( 2 π x 2 ) .

The domain Ω is uniformly divided by the triangular elements of T H with mesh size H and T h with mesh size h . J is uniformly divided so that τ is a small time step. Here, we fix the time step τ = 1.0 × 1 0 − 3 . The convergence rates in tables are computed as

rate = log ( E i ⁄ E i + 1 ) log ( h i ⁄ h i + 1 ) ,

where E i is the error corresponding to h i .

We use RT k ( k = 1 , 2 ) as the approximation space. For the two-grid method, we first solve for ( u H , p ˜ H , p H ) ∈ W H × V H × V H on the coarse grid using the nonlinear system (2.8)–(2.10), and then obtain ( U h , P ˜ h , P h ) ∈ W h × V h × V h on the fine grid using a linearized scheme (2.11)–(2.13) with h k + 1 = H 2 k + 1 ( k = 1 , 2 ), employing Newton’s iteration with a tolerance of 1 0 − 6 . The number of iterations depends on the specified tolerance during actual calculations. The computational data at t = 1 for our two-grid method are presented in Table 1, showing that ∣ ∣ u − U h ∣ ∣ , ∣ ∣ p ˜ − P ˜ h ∣ ∣ and ∣ ∣ p − P h ∣ ∣ are close O ( h k + 1 ) ( k = 1 , 2 ), which aligns with Theorem 4.2. To demonstrate both accuracy and efficiency of our two-grid method compared to EMFEM that directly employs Newton’s iteration in fine grid space to solve nonlinear problem (5.1), we present errors and computing time comparison between these methods in Table 2. From this table, it is observed that given the same accuracy, the two-grid method is more efficient than EMFEM regarding computational time. Furthermore, in Figure 1, we depict exact solution u at t = 1 and make comparisons with two-grid solution U h when h = 1 ⁄ 64 , τ = 1.0 × 1 0 − 3 , and t = 1 . From Figure 1, it is evident that there is no visual difference between exact solution u and two-grid solution U h . In Figure 2, we obtain the errors of ( U h , P ˜ h , P h ) in different time levels by our two-grid algorithm.

Table 1

Errors and convergence rate of the two-grid method with RT k ( k = 1 , 2 ) element

RT k ( H , h ) ∣ ∣ u ‒ U h ∣ ∣ Rate ∣ ∣ p ˜ ‒ P ˜ h ∣ ∣ Rate ∣ ∣ p ‒ P h ∣ ∣ Rate
k = 1 ( 1 ⁄ 4 , 1 ⁄ 8 ) 2.317 × 1 0 ‒ 2 — 5.650 × 1 0 ‒ 2 — 5.613 × 1 0 ‒ 2 —
( 1 ⁄ 9 , 1 ⁄ 27 ) 2.265 × 1 0 ‒ 3 1.91 5.574 × 1 0 ‒ 3 1.90 5.551 × 1 0 ‒ 3 1.90
( 1 ⁄ 16 , 1 ⁄ 64 ) 4.378 × 1 0 ‒ 4 1.90 1.079 × 1 0 ‒ 3 1.90 1.083 × 1 0 ‒ 3 1.89
( 1 ⁄ 25 , 1 ⁄ 125 ) 1.211 × 1 0 ‒ 4 1.92 3.025 × 1 0 ‒ 4 1.90 3.036 × 1 0 ‒ 4 1.90
( 1 ⁄ 36 , 1 ⁄ 216 ) 4.254 × 1 0 ‒ 5 1.91 1.063 × 1 0 ‒ 4 1.91 1.071 × 1 0 ‒ 4 1.90
k = 2 ( 1 ⁄ 8 , 1 ⁄ 32 ) 9.469 × 1 0 ‒ 4 — 3.382 × 1 0 ‒ 3 — 3.375 × 1 0 ‒ 3 —
( 1 ⁄ 27 , 1 ⁄ 243 ) 2.802 × 1 0 ‒ 6 2.87 9.991 × 1 0 ‒ 6 2.87 1.014 × 1 0 ‒ 5 2.86
Table 2

Errors and computing time of two-grid method and EMFEM with RT k ( k = 1 , 2 ) element

Two-grid method EMFEM
RT k ( H , h ) ‖ u ‒ U h ‖ Computing time (s) h ‖ u ‒ u h ‖ Computing time (s)
k = 1 ( 1 ⁄ 4 , 1 ⁄ 8 ) 2.317 × 1 0 ‒ 2 0.83 1/8 2.254 × 1 0 ‒ 2 0.97
( 1 ⁄ 9 , 1 ⁄ 27 ) 2.265 × 1 0 ‒ 3 7.25 1/27 2.223 × 1 0 ‒ 3 20.18
( 1 ⁄ 16 , 1 ⁄ 64 ) 4.378 × 1 0 ‒ 4 35.41 1/64 4.306 × 1 0 ‒ 4 112.05
( 1 ⁄ 25 , 1 ⁄ 125 ) 1.211 × 1 0 ‒ 4 119.58 1/125 1.198 × 1 0 ‒ 4 427.43
( 1 ⁄ 36 , 1 ⁄ 216 ) 4.254 × 1 0 ‒ 5 371.02 1/216 4.183 × 1 0 ‒ 5 1266.29
k = 2 ( 1 ⁄ 8 , 1 ⁄ 32 ) 9.469 × 1 0 ‒ 4 12.59 1/32 9.315 × 1 0 ‒ 4 37.83
( 1 ⁄ 27 , 1 ⁄ 243 ) 2.802 × 1 0 ‒ 6 384.67 1/243 2.744 × 1 0 ‒ 6 1409.55
Figure 1 Scalar solution. Left: the exact solution u u . Right: the numerical solution U h {{\mathcal{U}}}_{h} by two-grid method.
Figure 1

Scalar solution. Left: the exact solution u . Right: the numerical solution U h by two-grid method.

Figure 2 Errors of two-grid solution ( U h , P ˜ h , P h ) \left({{\mathcal{U}}}_{h},{\widetilde{{\mathcal{P}}}}_{h},{{\mathcal{P}}}_{h}) for t = 1 t=1 , τ = 1.0 × 1 0 ‒ 3 \tau =1.0\times 1{0}^{&#x2012;3} , and h = 1 ⁄ 64 h=1/64 .
Figure 2

Errors of two-grid solution ( U h , P ˜ h , P h ) for t = 1 , τ = 1.0 × 1 0 ‒ 3 , and h = 1 ⁄ 64 .

6 Conclusions

In this article, we present a two-grid expended mixed element discretization scheme for second-order hyperbolic equation. It is shown theoretically and numerically that two-grid method saving a large amount of computational time and without losing accuracy as long as the mesh sizes satisfy H 2 k + 1 = h k + 1 . The two-grid algorithm presented in this article involves only one Newton iteration on the fine grid; therefore, one more Newton iteration and correction technique can be considered to improve the error estimate. In the next work, we shall consider more efficient two-grid method for nonlinear hyperbolic equation and more complicated problems. Furthermore, inspired by [15,32], it is interesting to use new EMFEM to solve hyperbolic equation.

Acknowledgement

The author is grateful for the reviewer’s valuable comments that improved the manuscript.

  1. Funding information: This work was supported by the Science and Technology Plan Project of Hunan Province (2016TP1020), the “Double First-Class” Applied Characteristic Discipline in Hunan Province (Xiangjiaotong[2018]469), and the Excellent Youth Project of Hunan Education Department (22B0711).

  2. Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and prepared the manuscript.

  3. Conflict of interest: The author declares no conflicts of interest.

  4. Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

[1] D. M. Boore, Finite Difference Methods for Seismic Wave Propagation in Heterogeneous Materials, Academic Press, London, 1972. 10.1016/B978-0-12-460811-5.50006-4Search in Google Scholar

[2] L. M. Brekhovskihk, Waves in Layered Media, Academic Press, New York, 1980. Search in Google Scholar

[3] B. Lombard and J. Piraux, Numerical treatment of two-dimensional interfaces for acoustic and elastic waves, J. Comput. Phys. 195 (2004), 90–116, DOI: https://doi.org/10.1016/j.jcp.2003.09.024. 10.1016/j.jcp.2003.09.024Search in Google Scholar

[4] G. A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM J. Numer. Anal. 13 (1976), 564–576, DOI: https://doi.org/10.1137/0713048. 10.1137/0713048Search in Google Scholar

[5] T. Geveci, On the application of mixed finite element methods to the wave equation, Math. Model. Numer. Anal. 22 (1988), 243–250, DOI: https://doi.org/10.1051/m2an/1988220202431. 10.1051/m2an/1988220202431Search in Google Scholar

[6] L. Cowsar, T. Dupont, and M. F. Wheeler, A priori estimates for mixed finite element methods for the ware equation, Comput. Methods Appl. Mech. Eng. 82 (1990), 205–222, DOI: https://doi.org/10.1016/0045-7825(90)90165-I. 10.1016/0045-7825(90)90165-ISearch in Google Scholar

[7] Z. Zhou, W. Wang, and H. Chen, An H1-Galerkin expanded mixed finite element approximation of second-order nonlinear hyperbolic equations, Abstr. Appl. Anal. 2013 (2013), no. 1, 657952, DOI: https://doi.org/10.1155/2013/657952. 10.1155/2013/657952Search in Google Scholar

[8] C. J. Chen and W. Liu, A two-grid method for finite volume element approximations of second-order nonlinear hyperbolic equations, J. Comput. Appl. Math. 233 (2010), 2975–2984, DOI: https://doi.org/10.1016/j.cam.2009.11.043. 10.1016/j.cam.2009.11.043Search in Google Scholar

[9] Z. Chen, Expanded mixed finite element methods for linear second order elliptic problems (I), RAIRO: Math. Model. Numer. Anal. 32 (1998), 479–499, DOI: https://doi.org/10.1051/m2an/1998320404791. 10.1051/m2an/1998320404791Search in Google Scholar

[10] Z. Chen, Expanded mixed finite element methods for quasilinear second-order elliptic problems (II), RAIRO: Math. Model. Numer. Anal. 32 (1998), 501–520, DOI: https://doi.org/10.1051/m2an/1998320405011. 10.1051/m2an/1998320405011Search in Google Scholar

[11] T. Arbogast, M. F. Wheeler, and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal. 34 (1997), 828–852, DOI: https://doi.org/10.1137/S0036142994262585. 10.1137/S0036142994262585Search in Google Scholar

[12] C. S. Woodward and C. N. Dawson, Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modelling flow into variably saturated porous media, SIAM J. Numer. Anal. 37 (2000), 701–724, DOI: https://doi.org/10.1137/S0036142996311040. 10.1137/S0036142996311040Search in Google Scholar

[13] G. N. Gatica and N. Heuer, An expanded mixed finite element approach via a dual-dual formulation and the minimum residual method, J. Comput. Appl. Math. 132 (2001), 371–385, DOI: https://doi.org/10.1016/S0377-0427(00)00440-4. 10.1016/S0377-0427(00)00440-4Search in Google Scholar

[14] Y. Liu, Z. C. Fang, H. Li, S. He, and W. Gao, A new expanded mixed method for parabolic integro-differential equations, Appl. Math. Comput. 259 (2015), 600–613, DOI: https://doi.org/10.1016/j.amc.2015.02.081. 10.1016/j.amc.2015.02.081Search in Google Scholar

[15] N. Sharma, A. K. Pani, and K. K. Sharma, Expanded mixed FEM with lowest order RT elements for nonlinear and nonlocal parabolic problems, Adv. Comput. Math. 44 (2018), 1537–1571, DOI: https://doi.org/10.1007/s10444-018-9596-6. 10.1007/s10444-018-9596-6Search in Google Scholar

[16] J. Xu, A novel two-grid method for semilinear equations, SIAM J. Sci. Comput. 15 (1994), 231–237, DOI: https://doi.org/10.1137/0915016. 10.1137/0915016Search in Google Scholar

[17] J. Xu, Two-grid discretization techniques for linear and non-linear PDEs, SIAM J. Numer. Anal. 33 (1996), 1759–1777, DOI: https://doi.org/10.1137/S0036142992232949. 10.1137/S0036142992232949Search in Google Scholar

[18] C. Dawson and M. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, in: Domain Decomposition Methods in Scientific and Engineering Computing: Proc. of the Seventh International Conference on Domain Decomposition (University Park, PA), 1993, Contemporary Mathematics, pp. 191–203. 10.1090/conm/180/01971Search in Google Scholar

[19] L. Wu and M. B. Allen, A two-grid method for mixed finite-element solution of reaction-diffusion equations, Numer. Meth. Part. D. E. 15 (1999), no. 3, 317–332, DOI: https://doi.org/10.1002/(SICI)1098-2426(199905)15:3<317::AID-NUM4>3.0.CO;2-U. 10.1002/(SICI)1098-2426(199905)15:3<317::AID-NUM4>3.0.CO;2-USearch in Google Scholar

[20] Y. P. Chen, Y. Q. Huang, and D. H. Yu, A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations, Int. J. Numer. Methods Eng. 57 (2003), 193–209, DOI: https://doi.org/10.1002/nme.668. 10.1002/nme.668Search in Google Scholar

[21] Y. P. Chen, P. Luan, and Z. L. Lu, Analysis of two-grid methods for nonlinear parabolic equations by expanded mixed finite element methods, Adv. Appl. Math. Mech. 1 (2009), 830–844, DOI: https://doi.org/10.4208/aamm.09-m09S09. 10.4208/aamm.09-m09S09Search in Google Scholar

[22] Y. P. Chen, L. P. Chen, and X. C. Zhang, Two-grid method for nonlinear parabolic equations by expanded mixed finite element methods, Numer. Methods Partial Differential Equations 29 (2013), 1238–1256, DOI: https://doi.org/10.1002/num.21753. 10.1002/num.21753Search in Google Scholar

[23] H. Z. Hu, Y. P. Fu, and J. Zhou, Numerical solution of a miscible displacement problem with dispersion term using a two-grid mixed finite element approach, Numer. Algorithms 81 (2019), 879–914, DOI: https://doi.org/10.1007/s11075-018-0575-2. 10.1007/s11075-018-0575-2Search in Google Scholar

[24] H. Z. Hu, Two-grid method for compressible miscible displacement problem by mixed finite element methods and expanded mixed finite element method of characteristics, Numer. Algorithms 89 (2021), 611–636, DOI: https://doi.org/10.1007/s11075-021-01127-4. 10.1007/s11075-021-01127-4Search in Google Scholar

[25] Y. Liu, Y. W. Du, H. Li, J. C. Li, and S. He, A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative, Comput. Math. Appl. 70 (2015), 2474–2492, DOI: https://doi.org/10.1016/j.camwa.2015.09.012. 10.1016/j.camwa.2015.09.012Search in Google Scholar

[26] Y. Liu, Y. W. Du, H. Li, and J. F. Wang, A two-grid finite element approximation for a nonlinear time-fractional cable equation, Nonlinear Dynam. 85 (2016), 2535–2548, DOI: https://doi.org/10.1007/s11071-016-2843-9. 10.1007/s11071-016-2843-9Search in Google Scholar

[27] C. J. Chen, W. Liu, and C. J. Bi, A two-grid characteristic finite volume element method for semilinear advection-dominated diffusion equations, Numer. Methods Partial Differential Equations 29 (2013), 1543–1562, DOI: https://doi.org/10.1002/num.21766. 10.1002/num.21766Search in Google Scholar

[28] C. J. Chen, X. Y. Zhang, G. D Zhang, and Y. Y. Zhang, A two-grid finite element method for nonlinear parabolic integro-differential equations, Int. J. Comput. Math. 96 (2019), 2010–2023, DOI: https://doi.org/10.1080/00207160.2018.1548699. 10.1080/00207160.2018.1548699Search in Google Scholar

[29] C. J. Chen, H. Liu, X. C. Zheng, and H. Wang, A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection-diffusion equations, Comput. Math. Appl. 79 (2020), 2771–2783, DOI: https://doi.org/10.1016/j.camwa.2019.12.008. 10.1016/j.camwa.2019.12.008Search in Google Scholar

[30] P. A. Raviart and T. M. Thomas, A mixed finite element method for second order elliptic problems, in: I. Galligam, E. Magenes (Eds.), Mathematical Aspects of FEM, Lecture Notes in Mathematics, Vol. 96, Springer-Verlag, Berlin, 1977, pp. 292–315. 10.1007/BFb0064470Search in Google Scholar

[31] F. Brezzi, J. Douglas, and L. D. Marini, Two families of mixed elements for second order elliptic problems, Numer. Math. 88 (1985), 217–235, DOI: https://doi.org/10.1007/BF01389710. 10.1007/BF01389710Search in Google Scholar

[32] B. J. Ji, J. S. Zhang, Y. Yu, and Y. Yu, A new expanded mixed finite element method for Kirchhoff type parabolic equation, Numer. Algorithms 92 (2023), 2405–2432, DOI: https://doi.org/10.1007/s11075-022-01396-7. 10.1007/s11075-022-01396-7Search in Google Scholar
Received: 2023-09-23
Revised: 2024-04-25
Accepted: 2024-07-25
Published Online: 2024-08-27

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0048
Keywords for this article
hyperbolic equation; expanded mixed finite element method; two-grid method; error estimate
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2024-0048/html
Scroll to top button