Volume 25, Issue 2

In this paper we study approximations of a singularly perturbed system of two coupled reaction-diffusion equations, in one dimension, by using piecewise linear finite elements on graded meshes. When the parameters are of different magnitudes, the solution exhibits in general two distinct but overlapping boundary layers. We prove that, when the mesh grading parameter is appropriately chosen, optimal error estimates in a balanced norm for piecewise linear elements can be obtained. Supporting numerical results are also presented.

In [L. Bourhrara, A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes, J. Comput. Phys. 397 2019, Article ID 108801], a numerical scheme based on a combined spherical harmonics and discontinuous Galerkin finite element method for the resolution of the Boltzmann transport equation is proposed. One of its features is that a streamline weight is added to the test function to obtain the variational formulation. In the present paper, restricting our attention to the advective part of the Boltzmann equation, we prove the convergence and provide error estimates of this numerical scheme. To this end, the original variational formulation is restated in a broken functional space. The use of broken functional spaces enables to build a conforming approximation, that is the finite element space is a subspace of the broken functional space. The setting of a conforming approximation simplifies the numerical analysis, in particular the error estimates, for which a Céa’s type lemma and standard interpolation estimates are sufficient for our analysis. For our numerical scheme, based on ℙ k {\mathbb{P}^{k}} discontinuous Galerkin finite elements (in space) on a mesh of size h and a spherical harmonics approximation of order N (in the angular variable), the convergence rate is of order 𝒪 ⁢ ( N - t + h k ) {\mathcal{O}(N^{-t}+h^{k})} for a smooth solution which admits partial derivatives of order k + 1 {k+1} and t with respect to the spatial and angular variables, respectively. For k = 0 {k=0} (piecewise constant finite elements) we also obtain a convergence result of order 𝒪 ⁢ ( N - t + h 1 2 ) {\mathcal{O}(N^{-t}+h^{\frac{1}{2}})} . Numerical experiments in one, two and three dimensions are provided, showing a better convergence behavior for the L 2 {L^{2}} -norm, typically of one more order, 𝒪 ⁢ ( N - t + h k + 1 ) {\mathcal{O}(N^{-t}+h^{k+1})} .

In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L 2 \mathbf{L}^{2} and energy norms, the temperature approximation in L 2 L^{2} and energy norms and pressure approximation in L 2 L^{2} -norm for t > 0 t>0 . Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.

We consider nodal-based Lagrangian interpolations for the finite element approximation of the Maxwell eigenvalue problem. The first approach introduced is a standard Galerkin method on Powell–Sabin meshes, which has recently been shown to yield convergent approximations in two dimensions, whereas the other two are stabilized formulations that can be motivated by a variational multiscale approach. For the latter, a mixed formulation equivalent to the original problem is used, in which the operator has a saddle point structure. The Lagrange multiplier introduced to enforce the divergence constraint vanishes in an appropriate functional setting. The first stabilized method consists of an augmented formulation including a mesh dependent term that can be regarded as the Laplacian of the divergence constraint multiplier. The second formulation is based on orthogonal projections, which can be recast as a residual based stabilization technique. We rely on the classical spectral theory to analyze the approximating methods for the eigenproblem. The stability and convergence aspects are inherited from the associated source problems together with an assumption which is discussed numerically. We investigate the performance of the proposed formulations and provide some convergence results validating the theoretical ones for several benchmark tests, including ones with smooth and singular solutions.

A Cahn–Hilliard–Allen–Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem with respect to the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.

We present in this paper a rigorous error analysis of the vector penalty-projection method for solving the time-dependent incompressible Stokes equations with open boundary conditions on part of the boundary. First, we prove the stability of the scheme. Then we provide an error analysis for the second-order vector penalty-projection method which shows that the convergence rate of the error on the velocity and the pressure is of order 2 in l ∞ ⁢ ( L 2 ⁢ ( Ω ) ) l^{\infty}(\mathbf{L}^{2}(\Omega)) and l 2 ⁢ ( L 2 ⁢ ( Ω ) ) l^{2}(L^{2}(\Omega)) respectively. In addition, it is shown that the splitting errors of the method varies as O ⁢ ( ε ) \mathcal{O}(\varepsilon) , where 𝜀 is a penalty parameter chosen as small as desired. Several numerical tests in agreement with the theoretical results are presented. To the best of our knowledge, this paper provides the first rigorous proof of optimal error estimates for second-order splitting schemes with open boundary conditions.

In this paper, a second-order algorithm based on the spectral deferred correction method is constructed for the time-dependent natural convection problem, which allows one to automatically increase the accuracy of a first-order backward-Euler time-stepping method through using spectral integration on Gaussian quadrature nodes and constructing the corrections. A complete theoretical analysis is presented to prove that this algorithm is unconditionally stable and possesses second-order accuracy in time. Numerical examples are given to confirm the theoretical analysis and the effectiveness of our algorithm.

In this paper we formulate and analyze a space-time finite element method for the numerical simulation of rotating electric machines where the finite element mesh is fixed in a space-time domain. Based on the Babuška–Nečas theory we prove unique solvability both for the continuous variational formulation and for a standard Galerkin finite element discretization in the space-time domain. This approach allows for an adaptive resolution of the solution both in space and time, but it requires the solution of the overall system of algebraic equations. While the use of parallel solution algorithms seems to be mandatory, this also allows for a parallelization simultaneously in space and time. This approach is used for the eddy current approximation of the Maxwell equations which results in an elliptic-parabolic interface problem. Numerical results for linear and nonlinear constitutive material relations confirm the applicability and accuracy of the proposed approach.

The P2 BDM H-div finite element is enriched by five P 5 P_{5} divergence-free bubbles on each triangle. The resulting finite element remains H-div and is also H-curl nonconforming. Thus the new finite element combined with the discontinuous P 1 P_{1} element for pressure, is stabilizer-free and still divergence-free in solving the Stokes equations on triangular meshes. The stabilization can also be done by three, four, or six P 5 P_{5} divergence-free bubble-enrichment each triangle, shown both in theory and in computation. Numerical tests show the advantage of the divergence-free element over the P 2 P_{2} Taylor–Hood finite element.

The free longitudinal vibrations of a rod are described by a differential equation of the form ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 {(P(x)y\prime)^{\prime}+\lambda P(x)y(x)=0} , where P ⁢ ( x ) {P(x)} is the cross section area at point x and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form 𝐀 ⁢ Y = Λ ⁢ 𝐁 ⁢ Y {\mathbf{A}Y=\Lambda\mathbf{B}Y} , where 𝐀 {\mathbf{A}} and 𝐁 {\mathbf{B}} are Jacobi and diagonal matrices dependent to cross section P ⁢ ( x ) {P(x)} , respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section P ⁢ ( x ) {P(x)} by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is O ⁢ ( h 2 ) {O(h^{2})} .

The p -Laplacian problem - ∇ ⋅ ( ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u ) = f {-\nabla\cdot((\mu+|\nabla u|^{p-2})\nabla u)=f} is considered, where μ is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher order terms, to the error in a quasi-norm. The involved constants being independent of μ, the solution, the mesh size and aspect ratio. An adaptive algorithm is proposed and numerical results are presented when p = 3 {p=3} . From this model problem, we propose a simplified error estimator and use it in the framework of an industrial application, namely a nonlinear Navier–Stokes problem arising from aluminium electrolysis.

In this paper, optimal convergence for an adaptive finite element algorithm for elastoplasticity is considered. To this end, the proposed adaptive algorithm is established within the abstract framework of the axioms of adaptivity [C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 2014, 6, 1195–1253], which provides a specific proceeding to prove the optimal convergence of the scheme. The proceeding is based on verifying four axioms, which ensure the optimal convergence. The verification is done by using results from [C. Carstensen, A. Schröder and S. Wiedemann, An optimal adaptive finite element method for elastoplasticity, Numer. Math. 132 2016, 1, 131–154], which presents an alternative approach to optimality without explicitly relying on the axioms