Toward anisotropic mesh construction and error estimation in the finite element method

Directional, anisotropic features like layers in the solution of partial differential equations can be resolved favorably by using anisotropic finite element meshes. An adaptive algorithm for such meshes includes the ingredients Error estimation and Information extraction/Mesh refinement. Related articles on a posteriori error estimation on anisotropic meshes revealed that reliable error estimation requires an anisotropic mesh that is aligned with the anisotropic solution. To obtain anisotropic meshes the so‐called Hessian strategy is used, which provides information such as the stretching direction and stretching ratio of the anisotropic elements. This article combines the analysis of anisotropic information extraction/mesh refinement and error estimation (for several estimators). It shows that the Hessian strategy leads to well‐aligned anisotropic meshes and, consequently, reliable error estimation. The underlying heuristic assumptions are given in a stringent yet general form. Numerical examples strengthen the exposition. Hence the analysis provides further insight into a particular aspect of anisotropic error estimation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 625–648, 2002; DOI 10.1002/num.10023     

A posteriori error estimation for convection dominated problems on anisotropic meshes

A singularly perturbed convection–diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

A posteriori error estimation of the new mixed element schemes for second order elliptic problem on anisotropic meshes

This paper presents a posteriori residual error estimator for the new mixed element scheme for second order elliptic problem on anisotropic meshes. The reliability and efficiency of our estimator are established without any regularity assumption on the mesh.     

A posteriori H1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes

The paper deals with a singularly perturbed reaction diffusionmodel problem. The focus is on reliable a posteriori error estimatorsfor the H¹ seminorm that can be applied to anisotropic finiteelement meshes. A residual error estimator and a local problemerror estimator are proposed and rigorously analysed. They arelocally equivalent, and both bound the error reliably. Threemodifications of these estimators are introduced and discussed. Much attention is given to the performance of the error estimatorin numerical experiments. This helps to identify those estimatorsthat are suitable for practical applications.     

Robust a posteriori error estimation for the nonconforming Fortin-Soulie finite element approximation

We obtain a computable a posteriori error bound on the broken energy norm of the error in the Fortin-Soulie finite element approximation of a linear second order elliptic problem with variable permeability. This bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms. The estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error.

     

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.     

An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes

Summary. A new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes. All considerations carry over to anisotropic triangular meshes with minor changes only. The lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be a useful tool. With its help anisotropic interpolation estimates and subsequently the upper error bound are proven. Additionally it is pointed out how to treat Robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes. A numerical example supports the anisotropic error analysis. Received April 6, 1999 / Revised version received July 2, 1999 / Published online June 8, 2000     

Remarks on a posteriori error estimation for finite element solutions

We utilize the classical hypercircle method and the lowest-order Raviart–Thomas H(div)$H(\mathrm{div})$ element to obtain a posteriori error estimates of the _P1$P_1$ finite element solutions for 2D Poisson's equation. A few other estimation methods are also discussed for comparison. We give some theoretical and numerical results to see the effectiveness of the methods.     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.     

Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes

Summary. Both for the - and -norms, we prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes. We also show that, with a correct scaling, edge residuals yield a robust error estimator for a singularly perturbed reaction-diffusion equation. Received April 19, 1999 / Published online April 20, 2000     

A posteriori error estimation for the Stokes–Darcy coupled problem on anisotropic discretization

This paper presents an a posteriori error analysis for the stationary Stokes–Darcy coupled problem approximated by finite element methods on anisotropic meshes in or 3. Korn's inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to non‐conforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for non‐conforming case. With the obtained finite element solutions, the error estimators are constructed and based on the residual of model equations plus the stabilization terms. The lower error bound is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so‐called matching function is defined, and its discussion shows it to be useful tool. With its help, the upper error bound is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media. Copyright © 2016 John Wiley & Sons, Ltd.     

Gradient recovery type a posteriori error estimate for finite element approximation on non-uniform meshes

In this paper, we derive gradient recovery type a posteriori error estimate for the finite element approximation of elliptic equations. We show that a posteriori error estimate provide both upper and lower bounds for the discretization error on the non-uniform meshes. Moreover, it is proved that a posteriori error estimate is also asymptotically exact on the uniform meshes if the solution is smooth enough. The numerical results demonstrating the theoretical results are also presented in this paper.     

Nonconforming rotated Q_1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q1 element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

A locking-free anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem

This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.     

A posteriori error estimate for the mixed finite element method

A computable error bound for mixed finite element methods is established in the model case of the Poisson-problem to control the error in the H(div,) -norm. The reliable and efficient a posteriori error estimate applies, e.g., to Raviart-Thomas, Brezzi-Douglas-Marini, and Brezzi-Douglas-Fortin-Marini elements.

     

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

In this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm.     

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

An experimental survey of a posteriori Courant finite element error control for the Poisson equation

This comparison of some a posteriori error estimators aims at empirical evidence for a ranking of their performance for a Poisson model problem with conforming lowest order finite element discretizations. Modified residual-based error estimates compete with averaging techniques and two estimators based on local problem solving. Multiplicative constants are involved to achieve guaranteed upper and lower energy error bounds up to higher order terms. The optimal strategy combines various estimators.     

Polynomial preserving recovery for quadratic elements on anisotropic meshes

Polynomial preserving gradient recovery technique under anisotropic meshes is further studied for quadratic elements. The analysis is performed for highly anisotropic meshes where the aspect ratios of element sides are unbounded. When the mesh is adapted to the solution that has significant changes in one direction but very little, if any, in another direction, the recovered gradient can be superconvergent. The results further explain why recovery type error estimator is robust even under nonstandard and highly distorted meshes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A posteriori error estimate for finite volume element method of the parabolic equations

In this article, residual‐type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual‐type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual‐type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259–275, 2017