Element Approximations Of Elliptic Problems Research Articles

Adaptive Finite Element Approximations for Elliptic Problems using Regularized Forcing Data

We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is quasi-optimal in two-dimensional space and suboptimal in three-dimensional space. Numerical simulations are provided to confirm our findings.

Guaranteed contraction of adaptive inexact hp-refinement strategies with realistic stopping criteria

The purpose of this contribution it to theoretically analyze the adaptive refinement strategies for conforming hp-finite element approximations of elliptic problems proposed for exact algebraic solvers in (P. Daniel, A. Ern, I. Smears and M. Vohralk, Comput. Math. Appl. 76 (2018) 967–983.) and for inexact algebraic solvers in (P. Daniel, A. Ern, and M. Vohralk, Comput. Methods Appl. Mech. Eng. 359 (2020) 112607.). Both of these strategies are driven by guaranteed equilibrated flux energy error estimators. The employed hp-refinement criterion stems from solving two separate local residual problems posed only on the patches of elements around marked vertices selected by a bulk-chasing criterion. In the above references, we have derived a fully computable guaranteed bound on the ratio of the error on two successive steps of the hp-adaptive loop. Here, our focus is to prove that this ratio is uniformly smaller than one, and thus the convergence of the adaptive and adaptive inexact hp-refinement strategies. To be able to achieve this goal, we have to introduce some additional assumptions on the h- and p-refinements, namely an extension of the marked region, as well as a sufficient h- or p-refinement of each marked patch. We investigate two such strategies, where one ensures a polynomial-degree-robust guaranteed contraction. In the inexact case, a sufficiently precise stopping criterion for the algebraic solver is requested, but this criterion remains fully computable and also realistic in the sense that in our numerical experiments, it does not request the algebraic error to be excessively small in comparison with the total error.

An adaptive [formula omitted]-refinement strategy with inexact solvers and computable guaranteed bound on the error reduction factor

In this work we extend our recently proposed adaptive refinement strategy for hp-finite element approximations of elliptic problems by taking into account an inexact algebraic solver. Namely, on each level of refinement and on each iteration of an (arbitrary) iterative algebraic solver, we compute guaranteed a posteriori error bounds on the algebraic and the total errors in energy norm. The algebraic error is the difference between the inexact discrete solution obtained by an iterative algebraic solver and the (unavailable) exact discrete solution. On the other hand, the total error stands for the difference between the inexact discrete solution and the (unavailable) exact solution of the partial differential equation. For the algebraic error upper bound, we crucially exploit the whole nested hierarchy of hp-finite element spaces created by the adaptive algorithm, whereas the remaining parts of the total error upper and lower bounds are computed using the finest space only. These error bounds allow us to formulate adaptive stopping criteria for the algebraic solver ensuring that the algebraic error does not significantly contribute to the total error. Next, we use the total error bound to mark mesh vertices for refinement via Dörfler’s bulk-chasing criterion. On patches associated with marked vertices only, we solve two separate primal finite element problems with homogeneous Dirichlet (Neumann) boundary conditions, which serve to decide between h-, p-, or hp-refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the total errors of the inexact approximations between successive refinements (the error reduction factor), when the stopping criteria are satisfied. Finally, in a series of numerical experiments, we investigate the practicality of the proposed adaptive solver, the accuracy of our bound on the reduction factor, and show that exponential convergence rates are achieved even in the presence of an inexact algebraic solver.

An adaptive [formula omitted]-refinement strategy with computable guaranteed bound on the error reduction factor

We propose a new practical adaptive refinement strategy for hp-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of discrete local problems on vertex-based patches. The first type involves the solution on each patch of a mixed finite element problem with homogeneous Neumann boundary conditions, which leads to an H(div,Ω)-conforming equilibrated flux. This, in turn, yields a guaranteed upper bound on the error and serves to mark mesh vertices for refinement via Dörfler’s bulk-chasing criterion. The second type of local problems involves the solution, on patches associated with marked vertices only, of two separate primal finite element problems with homogeneous Dirichlet boundary conditions, which serve to decide between h-, p-, or hp-refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the errors between successive refinements (error reduction factor). In a series of numerical experiments featuring smooth and singular solutions, we study the performance of the proposed hp-adaptive strategy and observe exponential convergence rates. We also investigate the accuracy of our bound on the reduction factor by evaluating the ratio of the predicted reduction factor relative to the true error reduction, and we find that this ratio is in general quite close to the optimal value of one.

Super-convergence and post-processing for mixed finite element approximations of the wave equation

We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consider the subsequent time discretization by the Crank-Nicolson method and show that the analysis and the post-processing strategy can be generalized to the fully discrete schemes. Our proofs do not rely on duality arguments or inverse inequalities and the results therefore apply also for non-convex domains and non-uniform meshes.

Two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence

The purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (potential) variables, one for triangles and the other one for quadrilateral elements, to be used in discrete versions of the mixed finite element method for elliptic problems. They can be interpreted as enriched versions of BDFMk+1 spaces based on triangles, and of RTk spaces for quadrilateral elements. The new flux approximations are incremented with properly chosen internal shape functions (with vanishing normal components on the edges) of degree k+2, and matching primal functions of degree k+1 (higher than the border fluxes, which are kept of degree k). In all these cases, the divergence of the flux space coincide with the primal approximation space on the master element, producing stable simulations. Using static condensation, the global condensed system to be solved in the enriched cases has same dimension (and structure) of the original ones, which is proportional to the space dimension of the border fluxes for each element geometry. Measuring the errors withL2-norms, the enriched space configurations give higher convergence rate of order k+2 for the primal variable, while keeping the order k+1 for the flux. For affine meshes, the divergence errors have the same improved accuracy rate as for the error in the primal variable. For quadrilateral non-affine meshes, for instance trapezoidal elements, the divergence error has order k+1, one unit more than the order k occurring for RTk spaces on this kind of deformed meshes. This fact also holds for ABFk elements, but for them the potential order of accuracy does not improve, keeping order k+1.

Weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non‐matching meshes

SummaryIn this work, we propose a method to prescribe essential boundary conditions in the finite element approximation of elliptic problems when the boundary of the computational domain does not match with the element boundaries. The problems considered are the Poisson problem, the Stokes problem, and the Darcy problem, the latter both in the primal and in the dual formulation. The formulation proposed is of variational type. The key idea is to start with the variational form that defines the problem and treat the boundary condition as a constraint. The particular feature is that the Lagrange multiplier is not defined on the boundary where the essential condition needs to be prescribed but is taken as a certain trace of a field defined in the computational domain, either in all of it or just in a region surrounding the boundary. When approximated numerically, this may allow one to condense the DOFs of the new field and end up with a problem posed only in terms of the original unknowns. The nature of the field used to weakly impose boundary conditions depends on the problem being treated. For the Poisson problem, it is a flux; for the Stokes problem, a stress; for the Darcy problem in primal form, a velocity field; and for the Darcy problem in dual form, it is a potential. Copyright © 2014 John Wiley & Sons, Ltd.

Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

Abstract We propose an approach for goal-oriented error estimation in finite element approximations of second-order elliptic problems that combines the dual-weighted residual method and equilibrated-flux reconstruction methods for the primal and dual problems. The objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution. It is only assumed here that the discretization methods come with a priori error estimates and an equilibrated-flux reconstruction algorithm. A high-order discontinuous Galerkin (dG) method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes. One contribution of the paper is to show how the order of the dG method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method, a dG method, or a mixed Raviart–Thomas method is used for the solution of the primal problem. Numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size.

Finite Element Approximation of Elliptic Problems with Dirac Measure Terms in Weighted Spaces: Applications to One- and Three-dimensional Coupled Problems

In this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving Dirac measures, using weighted Sobolev spaces and weighted discrete norms. Our approach handles both the cases where the measure is simply a right-hand side and where it represents an additional term, i.e., solution-dependent, in the formulation of the problem. The main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains, where the coupling terms are seen as Dirac measures concentrated on the fractures. We first establish a decomposition lemma, which is our fundamental tool for the analysis of the considered problems in the nonstandard setting of weighted spaces. Then, we consider the stability of the Galerkin approximation with finite elements in weighted norms with uniform and graded meshes. We introduce a discrete decomposition lemma that extends the continuous one and allows us to derive discrete inf-sup conditions in weighted norms. Then, we focus on the convergence of the finite element method. Due to the lack of regularity, the convergence rates are suboptimal for uniform meshes; we show that using graded meshes, optimal rates are recovered. Our theoretical results are supported by several numerical experiments. Finally, we show how our theoretical results apply to certain coupled problems involving fluid flow in porous three-dimensional media with one-dimensional fractures that are found in the analysis of microvascular flows.

A posteriori error estimates for mixed finite element approximations of elliptic problems

We derive residual based a posteriori error estimates of the flux in L 2-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.

Two‐level preconditioning of discontinuous Galerkin approximations of second‐order elliptic equations

AbstractThis paper reviews some known and proposes some new preconditioning methods for a number of discontinuous Galerkin (or DG) finite element approximations for elliptic problems of second order. Nested hierarchy of meshes is generally assumed. Our approach utilizes a general two‐level scheme, where the finite element space for the DG method is decomposed into a subspace (viewed as an auxiliary or ‘coarse’ space), plus a correction which can be handled by a standard smoothing procedure. We consider three different auxiliary subspaces, namely, piecewise linear C0‐conforming functions, piecewise linear functions that are continuous at the centroids of the edges/faces (Crouzeix‐Raviart finite elements) and piecewise constant functions over the finite elements. To support the theoretical results, we also present numerical experiments for 3‐D model problem showing uniform convergence of the constructed methods. Copyright © 2006 John Wiley & Sons, Ltd.

Residual and hierarchicalaposteriorierror estimates for nonconforming mixed finite element methods

We analyze residual and hierarchical a posteriori error estimates for nonconforming finite element approximations of elliptic problems with variable coefficients. We consider a finite volume box scheme equivalent to a nonconforming mixed finite element method in a Petrov-Galerkin setting. We prove that all the estimators yield global upper and local lower bounds for the discretization error. Finally, we present results illustrating the efficiency of the estimators, for instance, in the simulation of Darcy flows through heterogeneous porous media.

Analysis of a subdomain‐based error estimator for finite element approximations of elliptic problems

AbstractIn this article we analyze a subdomain residual error estimator for finite element approximations of elliptic problems. It is obtained by solving local problems on patches of elements in weighted spaces and provides an upper bound on the energy norm of the error when the local problems are solved in sufficiently enriched discrete spaces. A guaranteed lower bound on the error is also derived by a simple postprocess of the solutions to the local problems. Numerical tests show very good effectivity indices for both the upper and lower bounds and a strong reliability of this estimator even for coarse meshes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 165–192, 2004

Domain decomposition for model heterogeneous anisotropic problem

Abstract The main focus of this paper is to suggest a domain decomposition method for finite element approximations of elliptic problems with anisotropic coefficients in domains consisting of anisotropic shape rectangles. The theorems on traces of functions from Sobolev spaces play an important role in studying boundary value problems of partial differential equations. These theorems are commonly used for a priori estimates of the stability with respect to boundary conditions, and also play very important role in constructing and investigating effective domain decomposition methods. The trace theorem for anisotropic rectangles with anisotropic grids is the main tool in this paper to construct domain decomposition preconditioners. Copyright © 2002 John Wiley & Sons, Ltd.

On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors

A method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal-oriented error estimation. In the latter, the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm. Goal-oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates. The performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one-dimensional problem.

Interior penalty preconditioners for mixed finite element approximations of elliptic problems

It is established that an interior penalty method applied to second-order elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method. This relation can be utilized in order to construct preconditioners for the discrete mixed system. As an example, a family of additive Schwarz preconditioners for these systems is constructed. Numerical examples which confirm the theoretical results are also presented.

Asymptotically Exact Error Estimators for Rectangular Finite Elements

This paper deals with a posteriors error estimates for finite element approximations of elliptic problems. Based on superconvergence results, a general approach is introduced to construct, under reasonable assumptions, asymptotically exact error estimators. Finally, several examples are presented and it is shown that other well-known estimators can be analyzed in this framework. The estimators introduced are of a local character and therefore inexpensive to compute.

On the asymptotic exactness of error estimators for linear triangular finite elements

This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution. One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.

Iterative refinement of finite element approximations for elliptic problems

Iterative refinement of finite element approximations for elliptic problems