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.     

Certified error bounds for uncertain elliptic equations

In many applications, partial differential equations depend on parameters which are only approximately known. Using tools from functional analysis and global optimization, methods are presented for obtaining certificates for rigorous and realistic error bounds on the solution of linear elliptic partial differential equations in arbitrary domains, either in an energy norm, or of key functionals of the solutions, given an approximate solution. Uncertainty in the parameters specifying the partial differential equations can be taken into account, either in a worst case setting, or given limited probabilistic information in terms of clouds.     

An iterative adaptive finite element method for elliptic eigenvalue problems

We consider the task of resolving accurately the n$n$th eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation (PDE), using an adaptive finite element method (FEM). Conventional adaptive FEM algorithms call a generalized eigensolver after each mesh refinement step. This is not practical in our situation since the generalized eigensolver needs to calculate n$n$ eigenpairs after each mesh refinement step, it can switch the order of eigenpairs, and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace. In order to circumvent these problems, we propose a novel adaptive algorithm that only calls a generalized eigensolver once at the beginning of the computation, and then employs an iterative method to pursue a selected eigenvalue–eigenfunction pair on a sequence of locally refined meshes. Both Picard’s and Newton’s variants of the iterative method are presented. The underlying partial differential equation (PDE) is discretized with higher-order finite elements (hp$hp$-FEM) but the algorithm also works for standard low-order FEM. The method is described and accompanied with theoretical analysis and numerical examples. Instructions on how to reproduce the results are provided.     

Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems

We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates.     

A unifying theory of a posteriori finite element error control

Summary Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm |||| of a linear functional of the formin the variable of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker of . As a consequence, any residual estimator that is a computable bound of |||| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.Supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.     

A unifying theory of a posteriori error control for nonconforming finite element methods

Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in Carstensen [Numer Math 100:617–637, 2005]. Therein, the key assumption is that the conforming first-order finite element space annulates the linear and bounded residual ℓ written . That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that . The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator with some elementary properties. It is conjectured that the more general hypothesis (H1)–(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and Navier–Lamé equations illustrate the presented unifying theory of a posteriori error control for NCFEM. Supported by DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the German Indian Project DST-DAAD (PPP-05). J. Hu was partially supported by National Science Foundation of China under Grant No.10601003.     

A framework for obtaining guaranteed error bounds for finite element approximations

We give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming, non-conforming, mixed and discontinuous finite element schemes. Whilst the details of the actual estimator are rather different for each particular scheme, there is nonetheless a common underlying structure at work in all cases. We aim to illustrate this structure by treating conforming, non-conforming and discontinuous finite element schemes in a single framework. In taking a rather general viewpoint, some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but, in return, we hope to allow the reader to ‘see the wood despite the trees’.     

Rate of convergence of regularization procedures and finite element approximations for the total variation flow

Summary We derive rates of convergence for regularization procedures (characterized by a parameter ) and finite element approximations of the total variation flow, which arises from image processing, geometric analysis and materials sciences. Practically useful error estimates, which depend on only in low polynomial orders, are established for the proposed fully discrete finite element approximations. As a result, scaling laws which relate mesh parameters to the regularization parameter are also obtained. Numerical experiments are provided to validate the theoretical results and show efficiency of the proposed numerical methods.     

Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations

This work concerns analysis and error estimates for optimal control problems related to implicit parabolic equations. The minimization of the tracking functional subject to implicit parabolic equations is examined. Existence of an optimal solution is proved and an optimality system of equations is derived. Semi-discrete (in space) error estimates for the finite element approximations of the optimality system are presented. These estimates are symmetric and applicable for higher-order discretizations. Finally, fully-discrete error estimates of arbitrarily high-order are presented based on a discontinuous Galerkin (in time) and conforming (in space) scheme. Two examples related to the Lagrangian moving mesh Galerkin formulation for the convection-diffusion equation are described.     

Adaptive multilevel finite element approximations of semilinear elliptic boundary value problems

Summary. In this paper we consider two aspects of the problem of designing efficient numerical methods for the approximation of semilinear boundary value problems. First we consider the use of two and multilevel algorithms for approximating the discrete solution. Secondly we consider adaptive mesh refinement based on feedback information from coarse level approximations. The algorithms are based on an a posteriori error estimate, where the error is estimated in terms of computable quantities only. The a posteriori error estimate is used for choosing appropriate spaces in the multilevel algorithms, mesh refinements, as a stopping criterion and finally it gives an estimate of the total error. Received April 8, 1997 / Revised version received July 27, 1998 / Published online September 24, 1999     

A uniqueness result for a semilinear elliptic problem: A computer-assisted proof

Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+up in Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ₁(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈[0,λ₁(Ω)), in the case Ω=²(0,1) and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ₁(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem.     

Stabilized discontinuous finite element approximations for Stokes equations

In this paper, we derive two stabilized discontinuous finite element formulations, symmetric and nonsymmetric, for the Stokes equations and the equations of the linear elasticity for almost incompressible materials. These methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity/displacements are imposed in a weak sense via Lagrange multipliers. For both methods, almost all reasonable pair of discontinuous finite element spaces can be used to approximate the velocity and the pressure. Optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in L²$L^2$ norm are obtained, as well as one in a mesh-dependent norm for the velocity in both symmetric and nonsymmetric formulations.     

Superconvergence of new mixed finite element spaces

In this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in [ETNA, Vol. 37, pp. 189-201, 2010]. Among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids, this space has the smallest unknowns. However, the scalar variable is only suboptimal in general; thus we have employed a post-processing technique for the scalar variable. As a byproduct, we have obtained a superconvergence on a rectangular grid. The superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory, but that of a scalar variable requires a new technique, especially for k=1. Numerical experiments are provided to support the theory.     

Besov regularity and new error estimates for finite volume approximations of the p-laplacian

Summary. In [1], we have constructed a family of finite volume schemes on rectangular meshes for the p-laplacian and we proved error estimates in case the exact solution lies in W^2,p. Actually, W^2,p is not a natural space for solutions of the p-laplacian in the case p>2. Indeed, for general L^p data it can be shown that the solution only belongs to the Besov space In this paper, we prove Besov kind a priori estimates on the approximate solution for any data in L^p. We then obtain new error estimates for such solutions in the case of uniform meshes     

An elliptic singularly perturbed problem with two parameters II: Robust finite element solution

In this paper we consider a singularly perturbed elliptic model problem with two small parameters posed on the unit square. The problem is solved numerically by the finite element method using piecewise linear or bilinear elements on a layer-adapted Shishkin mesh. We prove that method with bilinear elements is uniformly convergent in an energy norm. Numerical results confirm our theoretical analysis.     

Lower and upper solutions for elliptic problems in nonsmooth domains

In this paper we prove some existence results of semilinear Dirichlet problems in nonsmooth domains in presence of lower and upper solutions well-ordered or not. We first prove existence results in an abstract setting using degree theory. We secondly apply them for domains with conical points.     

Worst case scenario analysis for elliptic problems with uncertainty

This work studies linear elliptic problems under uncertainty. The major emphasis is on the deterministic treatment of such uncertainty. In particular, this work uses the Worst Scenario approach for the characterization of uncertainty on functional outputs (quantities of physical interest). Assuming that the input data belong to a given functional set, eventually infinitely dimensional, this work proposes numerical methods to approximate the corresponding uncertainty intervals for the quantities of interest. Numerical experiments illustrate the performance of the proposed methodology.     

Existence and comparison results for noncoercive and nonmonotone multivalued elliptic problems

We consider noncoercive quasilinear elliptic inclusions under multivalued flux boundary conditions involving multifunctions of Clarke’s generalized gradient. Our main goal is to provide existence and comparison results which are based on an appropriate generalization of the notions of subsolutions and supersolutions. Furthermore, compactness and extremality results of the solution set enclosed by an ordered pair of sub–supersolutions will be proved.     

Fourth-order compact finite difference method for fourth-order nonlinear elliptic boundary value problems

A compact finite difference method with non-isotropic mesh is proposed for a two-dimensional fourth-order nonlinear elliptic boundary value problem. The existence and uniqueness of its solutions are investigated by the method of upper and lower solutions, without any requirement of the monotonicity of the nonlinear term. Three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently. The convergence and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.     

Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.