Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction–diffusion problems

The bilinear finite element methods on appropriately graded meshes are considered both for solving singular and semisingular perturbation problems. In each case, the quasi-optimal order error estimates are proved in the ε-weighted H¹-norm uniformly in singular perturbation parameter ε, up to a logarithmic factor. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

Pointwise error estimates of the bilinear SDFEM on Shishkin meshes

A model singularly perturbed convection–diffusion problem in two space dimensions is considered. The problem is solved by a streamline diffusion finite element method (SDFEM) that uses piecewise bilinear finite elements on a Shishkin mesh. We prove that the method is convergent, independently of the diffusion parameter ε, with a pointwise accuracy of almost order 11/8 outside and inside the boundary layers. Numerical experiments support these theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Partial regularity and singular sets of solutions of higher order parabolic systems

In the present paper we provide a broad survey of the regularity theory for non-differentiable higher order parabolic systems of the type

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.     

Morley type non‐C0 nonconforming rectangular plate finite elements on anisotropic meshes

In this article, two Morley type non‐C⁰ nonconforming rectangular finite elements are discussed to numerically solve the fourth order plate bending problem under anisotropic meshes. The optimal anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches. Some numerical tests are given to confirm the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

New robust nonconforming finite elements of higher order

We show that existing quadrilateral nonconforming finite elements of higher order exhibit a reduction in the order of approximation if the sequence of meshes is still shape-regular but consists no longer of asymptotically affine equivalent mesh cells. We study second order nonconforming finite elements as members of a new family of higher order approaches which prevent this order reduction. We present a new approach based on the enrichment of the original polynomial space on the reference element by means of nonconforming cell bubble functions which can be removed at the end by static condensation. Optimal estimates of the approximation and consistency error are shown in the case of a Poisson problem which imply an optimal order of the discretization error. Moreover, we discuss the known nonparametric approach to prevent the order reduction in the case of higher order elements, where the basis functions are defined as polynomials on the original mesh cell. Regarding the efficient treatment of the resulting linear discrete systems, we analyze numerically the convergence of the corresponding geometrical multigrid solvers which are based on the canonical full order grid transfer operators. Based on several benchmark configurations, for scalar Poisson problems as well as for the incompressible Navier-Stokes equations (representing the desired application field of these nonconforming finite elements), we demonstrate the high numerical accuracy, flexibility and efficiency of the discussed new approaches which have been successfully implemented in the FeatFlow software (www.featflow.de). The presented results show that the proposed FEM-multigrid combinations (together with discontinuous pressure approximations) appear to be very advantageous candidates for efficient simulation tools, particularly for incompressible flow problems.     

Convergence of higher order finite volume schemes on irregular grids

We prove convergence to the entropy solution of a general class of higher order finite volume schemes on unstructured, irregular grids for multidimensional scalar conservation laws. Such grids allow for cells to become flat in the limit. We derive a new entropy inequality for higher order schemes built on Godunov's numerical flux. Our result implies convergence of suitably modified versions of MUSCL-type finite volume schemes, ENO schemes and the discontinuous Galerkin finite element method. Supported by Deutsche Forschungsgemeinschaft, SFB256.     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for semisingularly perturbed reaction–diffusion problems

The numerical approximation by a lower‐order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving semisingular perturbation problems. The quasi‐optimal‐order error estimates are proved in the ε‐weighted H¹‐norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε‐weighted H¹‐norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Local projection stabilisation for higher order discretisations of convection-diffusion problems on Shishkin meshes

We consider a singularly perturbed convection-diffusion equation on the unit square where the solution of the problem exhibits exponential boundary layers. In order to stabilise the discretisation, two techniques are combined: Shishkin meshes are used and the local projection method is applied. For arbitrary r≥2, the standard Q _r -element is enriched by just six additional functions leading to an element which contains the P _r+1. In the local projection norm, the difference between the solution of the stabilised discrete problem and an interpolant of the exact solution is of order uniformly in ε. Furthermore, it is shown that the method converges uniformly in ε of order in the global energy norm.      

Convergence and superconvergence analysis of a new quadratic Hermite-type triangular element on anisotropic meshes

A new quadratic Hermite-type triangular finite element is conceived to solve a class of two-dimensional second-order elliptic boundary value problems. Its error estimates on anisotropic meshes are developed. Furthermore, we verify that some conditions set to the meshes contribute to the proof of its superconvergence properties, which can improve the approximation results. Numerical examples are given to confirm our theoretical analysis.     

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     

Uniform superconvergence of a Galerkin finite element method on Shishkin‐type meshes

We consider a Galerkin finite element method that uses piecewise bilinears on a class of Shishkin‐type meshes for a model singularly perturbed convection‐diffusion problem on the unit square. The method is shown to be convergent, uniformly in the diffusion parameter ϵ, of almost second order in a discrete weighted energy norm. As a corollary, we derive global L₂‐norm error estimates and local L_∞‐norm estimates. Numerical experiments support our theoretical results. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16:426–440, 2000     

Superconvergence of conforming finite element for fourth‐order singularly perturbed problems of reaction diffusion type in 1D

We consider conforming finite element approximation of fourth‐order singularly perturbed problems of reaction diffusion type. We prove superconvergence of standard C¹ finite element method of degree p on a modified Shishkin mesh. In particular, a superconvergence error bound of in a discrete energy norm is established. The error bound is uniformly valid with respect to the singular perturbation parameter ?. Numerical tests indicate that the error estimate is sharp. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 550–566, 2014     

Convergence phenomena of \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $\mathcal{Q}_p$ \end{document}‐elements for convection–diffusion problems

We present a numerical study for singularly perturbed convection–diffusion problems using higher order Galerkin and streamline diffusion finite element method. We are especially interested in convergence and superconvergence properties with respect to different interpolation operators. For this we investigate pointwise interpolation and vertex‐edge‐cell interpolation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Automatic finite element formulation and assembly of hyperelastic higher order structural models

The formulation of higher order structural models and their discretization using the finite element method is difficult owing to their complexity, especially in the presence of nonlinearities. In this work a new algorithm for automating the formulation and assembly of hyperelastic higher-order structural finite elements is developed. A hierarchic series of kinematic models is proposed for modeling structures with special geometries and the algorithm is formulated to automate the study of this class of higher order structural models. The algorithm developed in this work sidesteps the need for an explicit derivation of the governing equations for the individual kinematic modes. Using a novel procedure involving a nodal degree-of-freedom based automatic assembly algorithm, automatic differentiation and higher dimensional quadrature, the relevant finite element matrices are directly computed from the variational statement of elasticity and the higher order kinematic model. Another significant feature of the proposed algorithm is that natural boundary conditions are implicitly handled for arbitrary higher order kinematic models. The validity algorithm is illustrated with examples involving linear elasticity and hyperelasticity.     

A CLASS OF SINGULAR PERTURBATIONS FOR SECOND ORDER QUASI－LINEAR BOUNDARY VALUE PROBLEMS ON INFINITE INTERVAL

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Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems

In this paper we prove the uniform convergence of the standard multigrid V-cycle algorithm with the Gauss-Seidel relaxation performed only on the new nodes and their "immediate" neighbors for discrete elliptic problems on the adaptively refined finite element meshes using the newest vertex bisection algorithm. The proof depends on sharp estimates on the relationship of local mesh sizes and a new stability estimate for the space decomposition based on the Scott-Zhang interpolation operator. Extensive numerical results are reported, which confirm the theoretical analysis.     

L 2 error estimates and superconvergence of the finite volume element methods on quadrilateral meshes

This paper is concerned with the finite volume element methods on quadrilateral mesh for second-order elliptic equation with variable coefficients. An error estimate in L ² norm is shown on the quadrilateral meshes consisting of h ²-parallelograms. Superconvergence of numerical solution is also derived in an average gradient norm on h ²-uniform quadrilateral meshes. Numerical examples confirm our theoretical conclusions.     

Convergence rates and interior estimates in homogenization of higher order elliptic systems

This paper is concerned with the quantitative homogenization of 2m-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp $O(\epsilon )$ convergence rate in $W^{m?1,p_0}$ with $p_0=\frac{2d}{d?1}$ in a bounded Lipschitz domain in ${\mathbb{R}}^d$ as well as the uniform large-scale interior $C^{m?1,1}$ estimate. With additional smoothness assumptions, the uniform interior $C^{m?1,1}$, $W^{m,p}$ and $C^{m?1,\alpha }$ estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.