A posteriori analysis of the spectral element discretization of a non linear heat equation

Mohamed Abdelwahed; Nejmeddine Chorfi

doi:10.1515/anona-2020-0140

Article Open Access

A posteriori analysis of the spectral element discretization of a non linear heat equation

Mohamed Abdelwahed and Nejmeddine Chorfi

Published/Copyright: August 7, 2020

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advances in Nonlinear Analysis Volume 10 Issue 1

Abstract

The paper deals with a posteriori analysis of the spectral element discretization of a non linear heat equation. The discretization is based on Euler’s backward scheme in time and spectral discretization in space. Residual error indicators related to the discretization in time and in space are defined. We prove that those indicators are upper and lower bounded by the error estimation.

Keywords: Non linear heat equation; Spectral elements discretization; Error indicators; A posteriori analysis

MSC 2010: 35J57; 65M70

1 Introduction

The a posteriori analysis technique presents a very efficient tool for the mesh adaptivity methods. These methods have been widely applied in the context of the finite element discretization (see [2, 4, 5, 14, 17, 18, 19, 20]). However, few works considered the discretization by the spectral method (see [1, 8, 10]).

This paper deals with the discretization of a non linear heat equation. We use Euler’s implicit scheme with respect to time and spectral element method with respect to space. The spectral element method consists in approaching the solution of a partial differential equation by polynomial function of high degree on each sub-domain of a decomposition.

This work is an extension of the results obtained by Bernardi et al. (see [8]) for a discretization based on the finite element method and Chorfi et al. (see [10]) for the case of a linear heat equation. Herein, we start by proving that the time semi discrete problem has a unique solution. We define two local families of residual error indicators (see [12]). A first family related to the time discretization and depends only on the discrete solution and the time step. The value of this indicator allows us to choose the next time step. A second family of error indicators concerns the spectral discretization and explicitly depends on the discrete solution and the data of the non linear heat equation. We prove the optimality of those indicators in the sense that their Hilbertian sum is upper and lower bounded by the error estimation with constants independent of the discrete parameter in space and time.

The paper is organized as follows:

In Section 2, we present the studied non linear heat equation. We prove the existence and uniqueness of the solution of the discrete time and full spectral problems.
Section 3 is related to the definition of error indicators deduced from the residue of the non linear heat equation. We prove the equivalence between the error estimation and the Hilbertian sum of those error indicators.

2 A time and space discrete problems

Let Ω be a bounded and simply connected domain of ℝ^d (d = 1, 2 or 3), ∂ Ω is its connected Lipschitz continuous boundary and T is a positive constant. We consider the non linear heat equation: Find φ solution of

∂φ∂t−div(λ(φ)∇φ)=fin Ω×]0,T[φ(x,t)=0on ∂Ω×]0,T[φ(x,0)=φ0in Ω. (2.1)

We suppose that λ is a function verifying

0<mλ≤λ(x)≤Mλ and |λ(x)−λ(y)|≤κλ|x−y|, (2.2)

f and φ_0 are given functions and φ is an unknown function.

In order to study the variational formulation of problem (2.1), we define the following spaces

H^s(Ω) s ≥ 0, is the Sobolev space provided with the norm ∥ . ∥_s,Ω and the semi-norm ∣ . ∣_s,Ω,
D(Ω) is the space of indefinitely differentiable functions with a compact support in Ω,
H0s (Ω) is the closure of D(Ω) in H^s(Ω) and H^−s(Ω) its dual space,
the scalar product in the space L²(Ω) is denoted by (.,.),
𝓒⁰(0, T; H^s(Ω)) is the space of continuous functions, with values in H^s(Ω),
L²(0, T; H^s(Ω)) is the space of square-integrable functions with values in H^s(Ω),
L²(0, T; H0s (Ω)) is the space of square-integrable functions with values in H0s (Ω).

Problem (2.1) is equivalent to the following variational formulation:

find φ ∈ 𝓒⁰(0, T, L²(Ω)) ∩ L²(0, T, H01 (Ω)) satisfying

φ(x,0)=φ0(x);x∈Ω (2.3)

and ∀ t ∈]0, T[, ∀ ψ ∈ H01 (Ω);

∂φ∂t(.,t),ψ+λ(φ(.,t))∇φ(.,t),∇ψ=<f(.,t),ψ>, (2.4)

where < .,. > is the duality product between H⁻¹(Ω) and H01 (Ω).

For a given data f in L²(0, T; L²(Ω)) and φ₀ in L²(Ω), the proof of existence of φ is based on the fixed point theorem. We refer to ([15], chap 7) and to ([3, 11]) for the detail of this proof.

We remark that by choosing ψ = φ(., t) in (2.4) and integrating between 0 and t, we obtain that the solution φ of problem (2.3)-(2.4) verifies the following stability condition: for t ∈ [0, T]

∥φ(0,t)∥L2(Ω)2+mλ∫0t∥∇φ(.,s)∥L2(Ω)d2ds≤∥φ0∥L2(Ω)2+1mλ∥f∥L2(0,t;L2(Ω))2.

We define the following norm:

for any φ ∈ L²(0, T, H01 (Ω)),

[[φ]]2(t)=∥φ∥L2(Ω)2+mλ∫0t∥∇φ(.,s)∥L2(Ω)d2ds. (2.5)

2.1 Time discretization

To define the time semi discrete problem, we begin by introducing a partition of the interval [0, T] in sub-intervals [t_k, t_k+1], for 1 ≤ k ≤ K, such that 0 = t₀ < t₁ < … < t_K = T. Let τ_k = t_k − t_k−1, τ = (τ₁, …, τ_K), |τ|=max1≤k≤K|τk| and

σk=max2≤k≤Kτkτk−1

the regularity parameter.

For the family (φ^k)_1≤k≤K = φ(., t_k), we associate the function φ_τ, defined on [0, T], affine on each sub-interval [t_k−1, t_k]; 1 ≤ k ≤ K, such that φ_τ(t_k) = φ(t_k), then

∀t∈[tk−1,tk],φτ(t)=φk−tk−tτk(φk−φk−1).

By applying the Euler implicit method, we deduce the following time semi-discrete problem:

φk−φk−1τk−div(λ(φk)∇φk)=fkin Ω,1≤k≤Kφk=0on ∂Ω,1≤k≤Kφ0=φ0in Ω, (2.6)

where f^k = f(., t^k).

Problem (2.6) is equivalent to the following variational formulation:

for f ∈ 𝓒⁰(0, T,H⁻¹(Ω)) and φ₀ ∈ L²(Ω), find (φ^k)_0≤k≤K ∈ L²(Ω) × ( H01 (Ω))^K such that for any 1 ≤ k ≤ K and ψ ∈ H01 (Ω):

(φk,ψ)+τk(λ(φk)∇φk,∇ψ)=(φk−1,ψ)+τk<fk,ψ>in Ω, (2.7)

φ0=φ0,inΩ. (2.8)

Theorem 2.1

For f ∈ 𝓒⁰(0, T; H⁻¹(Ω)) and φ₀ ∈ L²(Ω), problem (2.7)-(2.8) has a unique solution (φ^k)_0≤k≤K ∈ L²(Ω) × ( H01 (Ω))^K such that

∥φk∥L2(Ω)2+mλ∑j=1kτj∥∇φj∥L2(Ω)d2≤∥φ0∥L2(Ω)2+1mλ∑j=1kτj∥fj∥H−1(Ω)2. (2.9)

Proof 1

We begin by proving the existence of solution using Brouwer fixed point theorem ([13], chap 7).

For 1 ≤ k ≤ K, supposing φ^k−1 is known, we define the application ϕ^k, from H01 (Ω) into H01 (Ω), such that for φ^k ∈ H01 (Ω) and ψ ∈ H01 (Ω),

(ϕk(φk)/ψ)=(φk,ψ)+τk(λ(φk)∇φk,∇ψ)−(φk−1,ψ)−τk<fk,ψ>,

where (./.) is the scalar product in H¹(Ω).

Since λ is bounded, we conclude that ϕ^k is continuous in H01 (Ω) and verifies for all φ^k ∈ H01 (Ω) that

(ϕk(φk)/φk)≥∥φk∥L2(Ω)2+mλ|τ||φk|H01(Ω)2−∥φk−1∥L2(Ω)∥φk∥L2(Ω)−|τ|∥fk∥H−1(Ω)|φk|H01(Ω).

Then, we have

(ϕ(φk)/φk)≥min(1,|τ|mλ)∥φk∥H1(Ω)−(∥φk−1∥L2(Ω)+|τ|∥fk∥H−1(Ω))∥φk∥H1(Ω).

Then, (ϕ(φ^k)/φ^k) is non negative on the sphere of H01 (Ω) with radius

r=∥φk−1∥L2(Ω)+|τ|∥fk∥H−1(Ω)min(1,|τ|mλ).

Let (X_n)_n a decreasing sequence of sub-spaces of H¹(Ω) such that ∪n=0+∞ X_n is dense in H¹(Ω), ϕ^k remains continuous on X_n and verifies the non negative property in X_n. Following Brouwer’s fixed point theorem ([13], chap 7, corol. 1.2), there exists φnk in X_n and ∥φnk∥H1(Ω)≤r such that

∀θm∈H1(Ω)(ϕk(φnk)/θm)=0;m≤n. (2.10)

Equation (2.10) is written as:

for any θ_m ∈ X_m ⊂ X_n

(φnk,θm)+τk(λ(φnk)∇φnk,∇θm)=(φnk−1,θm)+τk(fk,θm). (2.11)

Since the sequence (φnk) is bounded in H¹(Ω), there exists a sub-sequence (φnpk) which weakly converges to φ^k in H¹(Ω). Consequently and according to the properties of the function λ (see (2.2)), we have:

for each θ_m in X_m

∫Ωλ(φnpk)∇φnpk∇θmdx=∫Ω∇φnpk(λ(φnpk)−λ(φk))∇θmdx+∫Ω∇φnpkλ(φk)∇(θm)dx.

We conclude that

limp→+∞∫Ωλ(φnpk)∇φnpk∇θmdx=∫Ωλ(φk)∇φk∇θmdx.

The convergence of the remaining terms in (2.11) is easy to prove since they are linear.

By the density of ∪m=0∞Xm in H¹(Ω), we deduce that φ^k is solution of problem

∀θ∈H01(Ω),(φk,θ)+τk(λ(φk)∇φk,∇θ)=(φk−1,θ)+τk(fk,θ).

Showing now the stability condition (2.9): Let ψ = φ^j in (2.7), then,

∥φj∥L2(Ω)+mλτj∥∇φj∥L2(Ω)d2≤∥φj−1∥L2(Ω)+τjmλ∥fj∥H−1(Ω).

Doing sum on j from 0 to k, we conclude the stability condition (2.9).□

Definition 2.2

We define the “local” norm on each ψ^k in H01 (Ω) by

[[ψk]]=(∥ψk∥L2(Ω)2+τk|ψk|H1(Ω)2)12, (2.12)

and the full time discrete norm for all (ψ^k)_0≤k≤K ∈ L²(Ω) × ( H01 (Ω))^K by

[[ψk]]k=∥ψ0∥L2(Ω)2+mλ∑j=1kτj∥∇ψj∥L2(Ω)d212. (2.13)

Lemma 2.1

For each (φ^k)_0≤k≤K in L²(Ω) × (H¹(Ω))^K, we have

14[[φk]]k2≤[[φτ]]2(tk)≤12(1+στ)[[φk]]k2+mλ2τ1∥∇φ0∥L2(Ω)d2. (2.14)

Proof 2

Let

αk=∫tk−1tk∥∇φτ(.,s)∥L2(Ω)d2andβk=τk∥∇φk∥L2(Ω)d2.

Following the definition of φ_τ, we deduce that

where (./.) is the scalar product in ℝ^d.

Then, we have

αk=τk3∥∇φτk∥L2(Ω)d2+∥∇φτk−1∥L2(Ω)d2+(∇φk,∇φk−1).

Using inequality ab≥−a24−b2, we deduce

αk≥τk4∥∇φτk∥L2(Ω)d2=14βk.

To show the other inequality, we use the property ab≤a22+b22. We obtain

αk≤τk2∥∇φτk∥L2(Ω)d2+∥∇φτk−1∥L2(Ω)d2. (2.15)

For k = 1, we keep inequality (2.15). For k ≥ 2 we use the following inequality

αk≤τk2∥∇φτk∥L2(Ω)d2+τk−12στ∥∇φτk−1∥L2(Ω)d2.

Doing the sum on k, we conclude (2.14).□

2.2 Spectral element discretization

Since the polynomials inverse inequalities are not optimal in dimension d ≥ 2, herinafter, we consider only the one dimensional case for the a posteriori analysis of the spectral element method applied to the non linear heat equation. We start by describing the discrete problem deduced from the problem (2.7)-(2.8).

Let Ω =] − 1, 1[, we perform a partition of Ω such that 1 = α₀ < α₁ < … < α_i < … < α_I = 1, where Ω_i =]α_i−1, α_i[; for 1 ≤ i ≤ I and h_i = α_i − α_i−1. Let N_i an integer greater than 2, associated to the sub-domain Ω_i, we define the discrete parameter

δ=((h1,N1),(h2,N2),…,(hI,NI)).

Let ξ0N<ξ1N<…<ξNN the zeros of the polynomial (1 − x²)L′_N(x), where N an integer greater than 2 and L_N the Legendre polynomial defined on Ω.

We recall the following Gauss-Lobatto quadrature formula:

∀ψ∈P2N−1(Λ);∫−11ψ(x)dx=∑j=0Nψ(ξjN)ρjN (2.16)

where ρjN, 0 ≤ j ≤ N, represent the weights.

Let ℙ_n(Ω) the space of polynomial of degree ≤ n. We define in L²(Ω) the discrete scalar product:

For any continuous functions φ and ψ on Ω

(φ,ψ)δ=∑i=1I∑j=0Nφ(ξjNi)ψ(ξjNi)ρjNi, (2.17)

where ξjNi=Fi−1(ξjN) and ρjNi=(αi−αi−1)ρjN,0≤j≤N, such that 𝓕_i is the bijection from Ω_i into Ω.

Let the space

Zδ=φδ∈H1(Ω);φδ/Ωi∈PNi(Ωi),1≤i≤I,

we recall the following property (see [7] for its proof)

∀φδ∈Zδ,∥φδ∥L2(Ω)2≤(φδ,φδ)δ≤3∥φδ∥L2(Ω)2. (2.18)

We consider i_δ the Lagrange interpolation operator such that for φ continuous on Ω̄_i; i_δ(φ)/_{Ω_i} ∈ ℙ_{N_i}(Ω_i) and for 0 ≤ j ≤ N_i

iδ(φ)/Ωi(ξjNi)=φ/Ωi(ξjNi). (2.19)

Let the discrete space

Xδ=φδ∈H01(Ω);φδ/Ωi∈PNi(Ωi);1≤i≤I).

Then, the full discrete problem is: find φδk in Zδ×Πk=1KXδ such that,

φδ0=iδ(φ0) in Ω (2.20)

∀ψδ∈Xδ,(φδk,ψδ)δ+τk(λδ(φδk)∇φδk,∇ψδ)δ=(φδk−1,ψδ)δ+τk(fk,ψδ)δ, (2.21)

where λ_δ is defined, for each φ continuous on Ω̄, by

λδ(φ)=minmax{iδ(λ(φ));mλ};Mλ. (2.22)

Theorem 2.2

For each 1 ≤ k ≤ K; we suppose that f^k ∈ H⁻¹(Ω), φ⁰ ∈ L²(Ω) and φδk−1 ∈ 𝕏_δ. Problem (2.20)-(2.21) has a unique solution φδk in 𝕏_δ verifying

[[φδk]]k2≤∥iδφ0∥L2(Ω)2+1mλ∑j=1kτj∥fj∥H−1(Ω)2.

The proof of the above theorem follows exactly the same idea as the proof of Theorem 2.1 by using Brouwer’s fixed point Theorem. We simply adjust the discrete norm to the continuous norm using inequality (2.18).

We refer to ([16], chap 13) for the a priori analysis of the finite element discretization of this type of problems when the triangulations are independent of time.

Remark 2.1

Calculating the nonlinear term (λδ(φδk)∇φδk,∇ψδ)δ using the quadrature formula (2.16) is made simpler by choosing a semi-linear λ such that λδ(φδk)∇φδk∇ψδ is a polynomial of degree ≤ 2N_i − 1 on each sub-domain Ω_i.

3 Error indicators, lower and upper bounds

This section deals with the definition of the two families of error indicators. The first indicator is related to time discretization and the second one to spectral element discretization. We prove the equivalence of those indicators with the error estimate.

3.1 The error indicators

The time error indicators are defined by analogy to our previous work in the linear case (see [10, 12]). For each k, 1 ≤ k ≤ K,

βk=(τk3)12∥λδ(φδk)12ddx(φδk−φδk−1)∥L2(Ω). (3.1)

We also define the local indicators, which can be computed explicitly as a function of the discrete solution:

For each k, 1 ≤ k ≤ K and each sub-interval Ω_i

ζk,i=Ni−1∥(iδfk−φδk−φδk−1τk+ddx(λδ(φδk)dφδkdx))(x−αi−1)12(αi−x)12∥L2(Ωi).

For technical reasons related to forthcoming demonstrations, we define the following discrete space:

Xδ−={φδ∈H01(Ω);λδ(φδ)φδ|Ωi∈PNi−1(Ωi),1≤i≤I}.

3.2 An upper bound for the error

Hereinafter, to upper-bound the error defined with the norm introduced in (2.5) by the errors indicators and the data function, we apply the triangular inequality:

[[φ−φδτ]](tk)≤[[φ−φτ]](tk)+[[φτ−φδτ]](tk). (3.2)

For the estimation of the error [[φ − φ_τ]](t_k), we consider π_τ the interpolating operator defined as follows:

For any function ψ continuous on [0, T], π_τψ is constant on each interval [t_k−1, t_k], 1 ≤ k ≤ K, equal to ψ(t_k).

Proposition 3.1

Suppose that the data function f belongs to 𝓒⁰(0, T; H⁻¹(Ω)) and the function φ₀ belongs to H01 (Ω). We assume that the solution (φ^k)_0≤k≤K of problem (2.6) satisfies, for p > 1,

sup0≤k≤K∥dφkdx∥Lp(Ω)≤y, (3.3)

where y is a constant depending only on the data f, φ₀ and λ. Then, there exists a positive constant C, depending only on T, m_λ, M_λ, κ_λ and y, such that the following a posteriori error estimate holds between the solution φ of problem (2.1) and the solution (φ^k)_0≤k≤K of problem (2.6), for all t_k, 1 ≤ k ≤ K,

[[φ−φτ]](tk)≤C((1+στ)12[[φτ−φδτ]](tk)+(∑m=1kβm2)12+∥f−πτf∥L2(0,tk;H−1(Ω))). (3.4)

Proof 3

By replacing φ = φ_τ in (2.4), we obtain, for any ψ ∈ H01 (Λ) and t ∈ [t_k−1, t_k],

(∂tφτ,ψ)+(λ(φτ)∂xφτ,∂xψ)=(φk−φk−1τk,ψ)+(λ(φτ)∂x(φτ)−λ(φk)∂x(φk),∂xψ)+(λ(φk)∂xφk.∂xψ).

Then, considering equation (2.6),

(∂tφτ,ψ)+(λ(φτ)∂xφτ,∂xψ)=(fk,ψ)+(λ(φτ)∂x(φτ)−λ(φk)∂x(φk),∂xψ). (3.5)

The difference between equation (3.5) and (2.4) results in:

(∂t(φ−φτ),ψ)+(λ(φ)∂x(φ)−λ(φτ)∂x(φτ),∂xψ)=(f−fk,ψ)+(λ(φk)∂x(φk)−λ(φτ)∂x(φτ),∂xψ). (3.6)

Since we have, for any u and v,

λ(u)∂x(u)−λ(v)∂x(v)=(λ(u)−λ(v))∂x(u)+λ(v)∂x(u−v) (3.7)

and considering ψ = (φ − φ_τ), we obtain:

(∂t(φ−φτ),(φ−φτ))+((λ(φ)−λ(φτ))∂xφ,∂x(φ−φτ))+(λ(φτ)∂x(φ−φτ),∂x(φ−φτ))=(f−fm,φ−φτ)+((λ(φm)−λ(φτ))∂xφm,∂x(φ−φτ))+(λ(φτ)∂x(φm−φτ),∂x(φ−φτ)).

After integrating between [t_m−1, t_m] and doing the sum on m, one can conclude from the properties of the function λ in (2.2) and the Hölder inequality for 1p+1p∗=12 the following:

12∥(φ−φτ)(tk)∥L2(Ω)2+mλ∫0tk∥∂x(φ−φτ)(s)∥L2(Ω)2ds≤∑m=1k(∫tm−1tm(f(s)−fm,(φ−φτ)(s))ds−mλ∫tm−1tm(∂x(φτ(s)−φm),∂x(φ−φτ)(s))ds+κλ∫tm−1tm∥φm−φτ(s)∥Lp∗(Ω)∥∂xφm∥Lp(Ω)∥∂x(φ−φτ)(s)∥L2(Ω)ds).

Let C_s the injection norm of the space H¹(Ω) in L^{p^*}(Ω). Then using inequality (3.3), we obtain:

12[[φ−φτ]]2(tk)≤∑m=1k(∫tm−1tm(f(s)−fm,(φ−φτ)(s))ds−mλ∫tm−1tm(∂x(φτ(s)−φm),∂x(φ−φτ)(s))ds+κλyCs∫tm−1tm∥∂x(φm−φτ(s))∥L2(Ω)∥∂x(φ−φτ)(s)∥L2(Ω)ds). (3.8)

Now, we proceed by evaluating the three terms in the right-hand side of the inequality (3.8).

From the definition of the operator π_τ, we deduce that:

|∫tm−1tm(f(s)−fm,(φ−φτ)(s))ds|≤(∫tm−1tm∥(f−πτf)(s)∥H−1(Ω)2ds)12(∫tm−1tm∥∂x(φ−φτ)(s)∥L2(Ω)2ds)12.

We notice also:

(∑m=1k∫tm−1tm∥∂x(φ−φτ)(s)∥L2(Ω)2ds)12≤[[φ−φτ]](tk).
Concerning the second and third terms, we proceed in the same way,

|∫tm−1tm(∂x(φτ(s)−φm),∂x(φ−φτ)(s))ds|≤(∫tm−1tm∥∂x(φτ(s)−φm)∥L2(Ω)2ds)12(∫tm−1tm∥∂x(φ−φτ)(s)∥L2(Ω)2ds)12.

Then, from the definition of φ_τ, we have:

∂x(φτ−φm)=(−tm−sτm)∂x(φm−φm−1), (3.9)

so,

∫tm−1tm∥∂x(φτ(s)−φm)∥L2(Ω)2ds=∥∂x(φm−φm−1)∥L2(Ω)2∫tm−1tm(s−tm)2τm2ds=τm3∥∂x(φm−φm−1)∥L2(Ω)2.

By the triangular inequality and (3.9), we obtain:

(∫tm−1tm∥∂x(φτ(s)−φm)∥L2(Ω)2ds)12≤(τm3)12∥∂x(φm−φδm)∥L2(Ω)+(τm3)12(1mλ)12∥λδ(φδm)12∂x(φδm−φδm−1)∥L2(Ω)+(τm3)12∥∂x(φm−1−φδm−1)∥L2(Ω).

Doing the sum on m, we conclude that there exists a positive constant C, depending only on T, m_λ, M_λ, κ_λ and y, such that:

∑m=1k∫tm−1tm∥∂x(φτ(s)−φm)∥L2(Ω)2ds≤C(∑m=1kβm2+∑m=1kτm3(∥∂x(φm−φδm)∥L2(Ω)2+∥∂x(φm−1−φδm−1)∥L2(Ω)2)). (3.10)

The property of norms equivalence (2.14) of lemma 2.1, yields that:

∑m=1kτm3(∥∂x(φm−φδm)∥L2(Ω)2+∥∂x(φm−1−φδm−1)∥L2(Ω)2)≤τ13∥∂x(φ0−φδ0)∥L2(Ω)2+∑m=1nτm3∥∂x(φm−φδm)∥L2(Ω)2+∑m=2kτm−1στ3∥∂x(φm−1−φδm−1)∥L2(Ω)2. (3.11)

So, we obtain:

∑m=1kτm3(∥∂x(φm−φδm)∥L2(Ω)2+∥∂x(φm−1−φδm−1)∥L2(Ω)2)≤C(1+στ)[[φτ−φτδ]]2(tk), (3.12)

where C is positive constant depending only on T, m_λ, M_λ, κ_λ and y. If we consider (3.8), (3.10), (3.11) and (3.12) together we conclude the result (3.4).□

To finish the global estimation, we have to bound the second term [[φ_τ − φ_τδ]](t_k), in the right-hand side of the inequality (3.2). The proof of this estimation requires the use of the orthogonal projection operator ΠN1,0 defined from H01 (Ω) into ℙ_N(Ω) ∩ H01 (Ω). See ([7], [9]) for more details about this operator and the proof of the next lemma.

Lemma 3.1

Let ψ ∈ H^p(] − 1, 1[) ∩ H01 (] − 1, 1[), p > 1. The following estimate holds

(∫−11(ψ−ΠN1,0ψ)2(x)(1−x2)−1dx)12≤CN−p∥ψ∥Hp(Ω),

where C is a positive constant independent of N.

Proposition 3.2

Suppose that the data function f belongs to 𝓒⁰(0, T; H⁻¹(Ω)) and the function φ₀ belongs to H01 (Ω). We assume that the solution (φ^k)_0≤k≤K of problem (2.7)-(2.8) satisfies the condition (3.3). Then, there exists a positive constant C, depending only on T, m_λ, M_λ, κ_λ and y, such that the following a posteriori error estimate holds between the solution (φ^k)_0≤k≤K of problem (2.7)-(2.8) and the solution (φδk)0≤k≤K of problem (2.20)-(2.21), for all t_k, 1 ≤ k ≤ K,

[[φk−φδk]]k≤C(∑m=1kτm∑i=1I(ζm,i2+∥fm−iδfm∥L2(Ωi)2))12+∥φ0−iδφ0∥L2(Ω). (3.13)

Proof 4

Let ψ = ψ_δ ∈ Xδ− in equation (2.7), then,

(φk,ψδ)+τk(λ(φk)∂xφk,∂xψδ)=(φk−1,ψδ)+τk(fk,ψδ).

Likewise, if we consider ψ_δ ∈ Xδ− in (2.21), the exactness of the quadrature formula (2.16) permits us to conclude,

(φδk,ψδ)+τk(λδ(φδk)∂xφδk,∂xψδ)=(φδk−1,ψδ)+τk(fk,ψδ)δ.

Then, for any ψ ∈ H01 (Ω), we have:

(φk−φδk,ψδ)+τk(λ(φk)∂x(φk−φδk),∂xψδ)=(φk−1−φδk−1,ψδ)+(φk−φδk,ψ−ψδ)+τk(λ(φk)∂xφk,∂x(ψ−ψδ))−τk(λδ(φδk)∂xφk,∂x(ψ−ψδ))−τk((λδ(φδk)−λ(φk))∂xφk,∂xψ)+τk(fk,ψδ)−τk(fk,ψδ)δ. (3.14)

Integrating by parts, we obtain:

(φk−φδk,ψ−ψδ)+τk(λ(φk)∂xφk,∂x(ψ−ψδ))−τk(λδ(φδk)∂xφδk,∂x(ψ−ψδ))=(φk−1−φδk−1,ψ−ψδ)+τk∑i=1I∫αi−1αi(fk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))(x)(ψ−ψδ)(x)dx−τk∑i=1I−1[∂xφδk](αi)(ψ−ψδ)(αi),

where [.] is the jump through the point α_i.

Using equality (3.14), we obtain:

(φk−φδk,ψ)+τk(λ(φk)∂xφk−λδ(φδk)∂xφδk,∂xψ)=(φk−1−φδk−1,ψ)+τk∑i=1I∫αi−1αi(fk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))(x)(ψ−ψδ)(x)dx−τk∑i=1I−1[∂xφδk](αi)(ψ−ψδ)(αi)+τk(fk,ψδ)−τk(fk,ψδ)δ.

The definition of the discrete product (2.17) yields:

And finally, using (2.19) we conclude,

Let z_δ the image of ψ by a local regularization operator (see [6] for the properties of such operator)

zδ=∑i=1IΠNi−11,0(ψ−ψ(αi−1)θi−1−ψ(αi)θi)+∑i=0Iψ(αi)θi,

where θ_i are continuous functions, affine on each Ω_i, equal to 1 in α_i and to 0 in the other nodes, (z_δ ∈ 𝕏_δ since ψ ∈ H01 (Ω)).

Considering ψ_δ = z_δ in equation (3.15), the jump term disappears in:

(φk−φδk,ψ)+τk(λ(φk)∂xφk−λδ(φδk)∂xφδk,∂xψ)=(φk−1−φδk−1,ψ)+τk∑i=1I∫αi−1αi(fk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))(x)(ψ−zδ)(x)dx+τk∑i=1I∫αi−1αi(fk−iδfk)(x)zδ(x)dx. (3.16)

Equation (3.16) can be written, by making appear the term (x−αi−1)12(αi−x)12, as follows:

(φk−φδk,ψ)+τk(λ(φk)∂xφk−λδ(φδk)∂xφδk,∂xψ)=(φk−1−φδk−1,ψ)+τk∑i=1I∫αi−1αi(fk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))(x−αi−1)12(αi−x)12(x−αi−1)−12(αi−x)−12(ψ−zδ)(x)dx+τk∑i=1I∫αi−1αi(fk−iδfk)(x)zδ(x)dx.

Thanks to Cauchy-Schwarz inequality, we have:

(φk−φδk,ψ)+τk(λ(φk)∂xφk−λδ(φδk)∂xφδk,∂xψ)≤(φk−1−φδk−1,ψ)+τk∑i=1I(∫αi−1αi(fk−iδfk)2(x)dx)12(∫αi−1aiψ2(x)dx)12+τk∑i=1I(∫αi−1αi(iδfk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))2(x)(x−αi−1)(αi−x)dx)12(∫αi−1αi(ψ−zδ)2(x)(x−αi−1)−1(αi−x)−1dx)12.

Then, using lemma 3.1, we obtain:

(φk−φδk,ψ)+τk(λ(φk)∂xφk−λδ(φδk)∂xφδk,∂xψ)≤(φk−1−φδk−1,ψ)+τk∑i=1Iζk,i∥∂xψ∥L2(Ωi)+τk∑i=1I∥fk−iδfk∥L2(Ωi)∥ψ∥L2(Ωi). (3.17)

Choosing ψ=φk−φδk and using the inequality ab≤a22+b22 in (3.17) leads:

∥φk−φδk∥L2(Ω)2+τk(λ(φk)∂xφk−λδ(φδk)∂xφδk,∂x(φk−φδk))≤∥φk−1−φδk−1∥L2(Ω)22+∥φk−φδk∥L2(Ω)22+τk∑i=1I(ζk,i+∥fk−iδfk∥L2(Ωi))∥∂x(φk−φδk)∥L2(Ωi)2.

Using Cauchy-Schwarz inequality, we have:

∥φk−φδk∥L2(Ω)22+τk(λ(φk)∂xφk−λδ(φδk)∂xφδk,∂x(φk−φδk))≤∥φk−1−φδk−1∥L2(Ω)22+τk(∑i=1I(ζk,i+∥fk−iδfk∥L2(Ωi))2)12(∑i=1I∥∂x(φk−φδk)∥L2(Ωi)2)12.

Applying again the inequality ab≤a22+b22, yields

∥φk−φδk∥L2(Ω)22+τk(λ(φk)∂xφk−λδ(φδk)∂xφδk,∂x(φk−φδk))≤∥φk−1−φδk−1∥L2(Ω)22+τk2∑i=1I(ζk,i+∥fk−iδfk∥L2(Ωi))+τk2∑i=1I∥∂x(φk−φδk)∥L2(Ωi)2.

Then, using equality (3.7) and the properties of the function λ in (2.2), (2.22) and (3.3), we deduce that, there exists a positive constant C only depending on T, m_λ, M_λ, κ_λ and y, such that:

∥φk−φδk∥L2(Ω)22+τk2∥∂x(φk−φδk)∥L2(Ω)2≤∥φk−1−φδk−1∥L2(Ω)22+Cτk∑i=1I(ζk,i2+∥fk−iδfk∥L2(Ωi)2).

We conclude (3.13) by doing the sum on k and applying lemma 2.1.□

The full a posteriori estimate subject of the following theorem is the result of propositions 3.1 and 3.2 combined together.

Theorem 3.1

Suppose that the data function f belongs to 𝓒⁰(0, T; H⁻¹(Ω)) and the function φ₀ belongs to H01 (Ω). We assume that the solution (φ^k)_0≤k≤K of problem (2.7)-(2.8) satisfies the condition (3.3). Then, there exists a positive constant C depending only on T, m_λ, M_λ, κ_λ and y, such that the following a posteriori error holds between the solution φ of problem (2.1) and the solution (φδk)0≤k≤K of problem (2.21)-(2.20), for all t_k, 1 ≤ k ≤ K,

[[φ−φδτ]](tk)≤C(∑m=1k(βm2+τm∑i=1I(ζm,i2+∥fm−iδfm∥L2(Ωi)2)))12+(∥φ0−iδφ0∥L2(Ω)+∥f−πτf∥L2(0,tk,H−1(Ω))).

3.3 An upper bound for the error indicators

In this section, we will focus on the upper bound of the error indicators β_k and ζ_k,i according to the error estimate.

Proposition 3.3

Assume that the data function f belongs to 𝓒⁰(0, T; H⁻¹(Ω)) and the function φ₀ belongs to H01 (Ω). We assume also the solution (φ^k)_0≤k≤K of problem (2.7)-(2.8), such that ∂_x φ^k ∈ L^p(Ω), p > 1, satisfies condition (3.3). Then, there exists a positive constant C depending only on T, m_λ, M_λ, κ_λ, ∣τ∣ and y, such that the following estimate holds for the indicator β_k, 1 ≤ k ≤ K :

βk≤C([[φk−φδk]]+(στ)12[[φk−1−φδk−1]]+(τk)12∥f−πτf∥L2(tk−1,tk;H−1(Ω))+(τk)12(∥∂t(φ−φτ)∥L2(tk−1,tk;H−1(Ω))+∥∂x(φ−φτ)∥L2(tk−1,tk;L2(Ω)))). (3.18)

Proof 5

From the expression of the error indicator in (3.1) and by the triangular inequality, we obtain:

βk≤(τk3)12(∥λδ(φδk)12∂x(φδk−φk)∥L2(Ω)+∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω)+∥λδ(φδk)12∂x(φk−1−φδk−1)∥L2(Ω)).

Using the definition of the local norm in (2.12), we conclude:

(τk3)12∥λδ(φδk)12∂x(φδk−φk)∥L2(Ω)≤mλ[[φk−φδk]]

and

(τk3)12∥λδ(φδk)12∂x(φk−1−φδk−1∥L2(Ω)≤(στ)12mλ[[φk−1−φδk−1]].

In order to estimate the term ∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω), we consider ψ = φ^k − φ^k−1 in (3.6), (3.7) and integrate between t_k−1, t_k,

∫tk−1tk(∂t(φ−φτ)(t),φk−φk−1)ds+∫tk−1tk(λ(φ)∂x(φ−φτ)(t),∂x(φk−φk−1))ds=∫tk−1tk(f(s)−fk,φk−φk−1)ds−∫tk−1tk(λδ(φδk)∂x(φτ(s)−φk,∂x(φk−φk−1))ds+∫tk−1tk((λ(φ)−λδ(φδk))∂xφτ(s),∂x(φk−φk−1))ds.

When we apply (3.9),

−∫tk−1tk(λδ(φδk)∂x(φτ(s)−φk),∂x(φk−φk−1))ds=∫tk−1tktk−sτk∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω)2ds=∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω)2∫tk−1tktk−sτkds=τk2∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω)2,

we derive:

τk2∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω)2=∫tk−1tk(∂t(φ−φτ)(t),φk−φk−1)ds+∫tk−1tk(λ(φ)∂x(φ−φτ)(t),∂x(φk−φk−1))ds−∫tk−1tk(f(s)−fk,φk−φk−1)ds+∫tk−1tk((λ(φ)−λδ(φδk))∂xφτ(s),∂x(φk−φk−1))ds. (3.19)

The first term in the right-hand side of the equation (3.19) is obviously bounded as follows (note that this requires that φ⁰ = φ₀ ∈ H01 (Ω)):

|∫tk−1tk(∂t(φ−φτ)(t),φk−φk−1)ds|≤(τkmλ)12∥∂t(φ−φτ)∥L2(tk−1,tk;H−1(Ω))∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω).

We also have:

|∫tk−1tk(λ(φ)∂x(φ−φτ)(t),∂x(φk−φk−1))|≤(1+τkmλ)12τk12∥λ(φ)12∂x(φ−φτ)∥L2(tk−1,tk;L2(Ω))∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω),

and

|∫tk−1tk(f(s)−fk,φk−φk−1)ds|≤(τkmλ)12∥∂t(f−πτf)∥L2(tk−1,tk;H−1(Ω))∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω).

The last term in the right-hand side of the equation (3.19) is estimated in the same manner as in the inequality (3.8).

|∫tk−1tk((λ(φ)−λδ(φδk))∂xφτ(s),∂x(φk−φk−1))ds|≤C∥λ(φ)12∂xφτ∥L2(tk−1,tk;L2(Ω))∥λδ(φδk)12∂x(φk−φk−1)∥L2(Ω),

where C is a positive constant depending only on T, m_λ, M_λ, κ_λ, ∣τ∣ and y.

We use (2.13) and (2.14) to evaluate the norm of φ_τ. Finally, we conclude the estimation (3.18) by inserting all these estimates into (3.19).□

For the following, we will be interested to upper bound the error indicator ζ_k,i, then we need to introduce the following lemma (see [6] for its proof).

Lemma 3.2

For any ψ_N belongs to the polynomial space ℙ_N(Ω), the inverse inequalities hold

∫−11ψN′2(x)(1−x2)2dx≤CN2∫−11ψN2(x)(1−x2)dx (3.20)

and

∫−11ψN2(x)dx≤CN2∫−11ψN2(x)(1−x2)dx, (3.21)

where C is a positive constant independent of N.

Proposition 3.4

Assume that the data function f belongs to 𝓒⁰(0, T; H⁻¹(Ω)) and the function φ₀ belongs to H01 (Ω). We assume also the solution (φ^k)_0≤k≤K of problem (2.7)-(2.8), such that ∂_x φ^k ∈ L^p(Ω), p > 1, satisfies the condition (3.3). Then, there exists a positive constant C depending only on T, m_λ, M_λ, κ_λ, ∣τ∣ and y such that the following estimate holds for the indicator ζ_k,i, 1 ≤ k ≤ K, 1 ≤ i ≤ I,

ζk,i≤C(∥∂x(φk−φδk)∥L2(Ωi)+∥(φk−φδk)−(φk−1−φδk−1)τk∥H−1(Ωi)+Ni−1hi∥fk−iδfk∥L2(Ωi)). (3.22)

Proof 6

Choosing in equality (3.15), ψ_δ = 0 and

ψ=(iδfk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))(x−αi−1)(αi−x)inΩi,ψ=0inΩ∖Ωi,

leads

∫αi−1αi(iδfk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))2(x)(x−αi−1)(αi−x)dx=(λ(φk)∂x(φk−φδk),∂xψ)−∫αi−1αi(fk−iδfk)(x)ψ(x)dx+((φk−φδk)−(φk−1−φδk−1)τk,ψ)+((λ(φk)−λδ(φδk))∂x(φδk),∂xψ).

Bounding the terms in the right-hand side follows the same techniques as in the previous proof. Thanks to Cauchy schwarz and the Hölder inequalities (1p+1p∗=12), we obtain

∫αi−1αi(iδfk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))2(x)(x−αi−1)(αi−x)dx≤∥(φk−φδk)−(φk−1−φδk−1)τk∥H−1(Ωi)∥∂xψ∥L2(Ωi)+∥∂x(φk−φδk)∥L2(Ωi)∥∂xψ∥L2(Ωi)+∥fk−iδfk∥L2(Ωi)∥ψ∥L2(Ωi)+∥λ(φk)−λδ(φδk)∥Lp∗(Ωi)∥∂x(φδk)∥Lp(Ωi)∥∂xψ∥L2(Ωi). (3.23)

Applying the formula (a + b)² ≤ 2a² + 2b², we have

∥∂xψ∥L2(Ωi)≤2∫αi−1αi(∂x(iδfk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk)))2(x)(x−αi−1)2(αi−x)2dx+2∫αi−1αi(iδfk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))2(αi+αi−1−2x)2dx.

Using the two inverse inequalities (3.20) and (3.21) of lemma 3.2, we obtain

∥∂xψ∥L2(Ωi)≤CNi∥(iδfk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))(x−αi−1)12(αi−x)12∥L2(Ωi) (3.24)

and

∥ψ∥L2(Ωk)≤Chi∥(iδfk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))(x−αi−1)12(αi−x)12∥L2(Ωi). (3.25)

The last term in the right-hand side of inequality (3.23) is bounded in the same way as in equation (3.8). Then, there exists a positive constant C depending only on T, m_λ, M_λ, κ_λ, ∣τ∣ and y, such that

∥λ(φk)−λδ(φδk)∥Lp∗(Ωi)∥∂x(φδk)∥Lp(Ωi)∥∂xψ∥L2(Ωi)≤C∥∂x(φk−φδk)∥L2(Ωi)∥∂xψ∥L2(Ωi). (3.26)

Finally, by inserting (3.24), (3.25) and (3.26) in (3.23), simplifying by ∥(iδfk−φδk−φδk−1τk+∂x(λδ(φδk)∂xφδk))(x−αi−1)12(αi−x)12∥L2(Ωi) and multiplying by Ni−1, we derive the inequality (3.22).□

Conclusion

The a posteriori analysis of the discretization of a partial differential equations is a very efficient tool for mesh adaptivity. In this paper, we were interested in the a posteriori analysis of the discretization of the non linear heat equation by the spectral element method. We constructed two residual type of indicators and we proved their optimal upper and lower error bounds.

Acknowledgements

Researchers Supporting Project number (RSP-2020/153), King Saud University, Riyadh, Saudi Arabia.

References

[1] M. Abdelwahed and N. Chorfi, On the convergence analysis of a time dependent elliptic equation with discontinuous coefficients, Adv. Nonlinear Anal. 2020; 9: 1145-1160.10.1515/anona-2020-0043Search in Google Scholar

[2] M. Ainsworth and J.T. Oden, Aprocedure for a posteriori error estimation for h-p finite element methods, Comp. Methods in Applied Mech. and Eng. 101 (1992), 73-96.10.1016/0045-7825(92)90016-DSearch in Google Scholar

[3] H.W. Alt and S. Luckhaus, Quasilinear elliptic parabolic differential equations, Math. Z. 183 (1983), 311-341.10.1007/BF01176474Search in Google Scholar

[4] I. Babuška and W.C. Rheinboldt, A posteriori error estimates for the finite element method, Int. J. Numer. Methods Eng. 12 (1978), 1597-1645.10.1002/nme.1620121010Search in Google Scholar

[5] A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations, Math. Comput. 74 (2005), 1117-1138.10.1090/S0025-5718-04-01697-7Search in Google Scholar

[6] C. Bernardi, Indicateurs ďerreur en h − N version des éléments spectraux, RAIRO Modélisation mathématique et analyse numérique 30 (1996), no 1, 1-38.10.1051/m2an/1996300100011Search in Google Scholar

[7] C. Bernardi and Y. Maday, Approximation spectrales de problèmes aux limites elliptiques, Springer-Verlag, Berlin (1992).Search in Google Scholar

[8] C. Bernardi, B. étivet and R. Verfürth, Analyse numérique ďindicateurs ďerreur, in Maillage et adaptation, P.-L. George ed., Hermès (2001), 251-278.Search in Google Scholar

[9] C. Bernardi and N. Chorfi, Spectral discretization of the vorticity velocity and pressure formulation of the Stokes problem, SIAM J. Numer. Anal. 44 (2006), 826-850.10.1137/050622687Search in Google Scholar

[10] N. Chorfi, M. Abdelwahed and I. Ben Omrane, A posteriori analysis of the spectral element discretization of heat equation, An. Şt. Univ. Ovidius Constanţa, 22 (2014), no 3, 13-35.10.2478/auom-2014-0047Search in Google Scholar

[11] N. Chorfi, M. Abdelwahed and L. C. Berselli, On the analysis of a geometrically selective turbulence model, Adv. Nonlinear Anal. 2020; 9: 1402-1419.10.1515/anona-2020-0057Search in Google Scholar

[12] C. Johnson, Y.-Y. Nie and V. Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal. 27 (1990), 277-291.10.1137/0727019Search in Google Scholar

[13] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986.10.1007/978-3-642-61623-5Search in Google Scholar

[14] J.T. Oden, L. Demkowicz, W. Rachowicz and T.A. Westermann, Toward a universal h-p adaptive finite element strategy, Part II: a posteriori error estimation, Comp. Methods Appl. Mech. and Eng. 77 (1989), 113-180.10.1016/0045-7825(89)90130-8Search in Google Scholar

[15] J.-L.Lions, Equations différentielles opérationnelles et problèmes aux limites, Springer (1961).10.1007/978-3-662-25839-2Search in Google Scholar

[16] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics 25, Springer (1997).10.1007/978-3-662-03359-3Search in Google Scholar

[17] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley et Teubner (1996).Search in Google Scholar

[18] R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math. 50 (1994), 67-83.10.1016/0377-0427(94)90290-9Search in Google Scholar

[19] R. Verfürth, A posteriori error estimators and adaptive mesh-refinement techniques for the Navier-Stokes equations, in Incomputational Fluid Dynamics, Trends and Advances, M.D. Gunzburger and R.A. Nicolaides, Cambridge University Press (1993), 447-477.10.1017/CBO9780511574856.015Search in Google Scholar

[20] R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math.Comput. 62 (1994), 445-475.10.1090/S0025-5718-1994-1213837-1Search in Google Scholar

Received: 2020-05-27

Accepted: 2020-06-16

Published Online: 2020-08-07

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

Articles in the same Issue

https://doi.org/10.1515/anona-2020-0140

Keywords for this article

Non linear heat equation; Spectral elements discretization; Error indicators; A posteriori analysis

Creative Commons

BY 4.0