Residual-free bubbles for advection-diffusion problems: the general error analysis

Summary. We develop the general a priori error analysis of residual-free bubble finite element approximations to non-self-adjoint elliptic problems of the form subject to homogeneous Dirichlet boundary condition, where A is a symmetric second-order elliptic operator, C is a skew-symmetric first-order differential operator, and is a positive parameter. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree . Received October 1, 1998/ Revised version received April 6, 1999 / Published online January 27, 2000     

Moving finite element methods by use of space–time elements: I. Scalar problems

This article deals with moving finite element methods by use of the time-discontinuous Galerkin formulation in combination with oriented space–time meshes. A principle for mesh orientation in space–time based on minimization of the residual, related to adaptive error control via an a posteriori error estimate, is presented. The relation to Miller's moving finite element method is discussed. The article deals with scalar problems; systems will be treated in a companion article. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:251–262, 1998     

Pointwise localized error estimates for parabolic finite element equations

Summary. We derive pointwise weighted error estimates for a semidiscrete finite element method applied to parabolic equations. The results extend those obtained by A.H. Schatz for stationary elliptic problems. In particular, they show that the error is more localized for higher order elements. Mathematics Subject Classification (2000): 65N30     

and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.     

Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

The subject of the paper is the derivation of error estimates for the combined finite volume-finite element method used for the numerical solution of nonstationary nonlinear convection-diffusion problems. Here we analyze the combination of barycentric finite volumes associated with sides of triangulation with the piecewise linear nonconforming Crouzeix-Raviart finite elements. Under some assumptions on the regularity of the exact solution, the L ²(L ²) and L ²(H ¹) error estimates are established. At the end of the paper, some computational results are presented demonstrating the application of the method to the solution of viscous gas flow.     

Penalty finite element approximations for the Stokes equations by L2 projection

This paper considers the penalty finite element method for the Stokes equations, based on some stable finite elements space pair (X_h, M_h) that do satisfy the discrete inf–sup condition. Theoretical results show that the penalty error converges as fast as one should expect from the order of the elements. Moreover, the penalty finite element method by L² projection can improve the penalty error estimates. Finally, we confirm these results by a series of numerical experiments. Copyright © 2008 John Wiley & Sons, Ltd.     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

Error analysis of a fully discrete projection method for magnetohydrodynamic system

In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi-implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H ( curl ) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.     

Numerical analysis of the primal problem of elastoplasticity with hardening

Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening. Received May 7, 1997 / Revised version received August 31, 1998     

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.     

An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics

We consider the coupling of dual‐mixed finite elements and boundary elements to solve a mixed Dirichlet–Neumann problem of plane elasticity. We derive an a‐posteriori error estimate that is based on the solution of local Dirichlet problems and on a residual term defined on the coupling interface. The general error estimate does not make use of any special finite element or boundary element spaces. Here the residual term is given in a negative order Sobolev norm. In practical applications, where a certain boundary element subspace is used, this norm can be estimated by weighted local L²‐norms. Copyright © 2001 John Wiley & Sons, Ltd.     

Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis

A preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed. The proposed preconditioner is of block triangular form. The aim of this article is to show that a rigorous convergence analysis can be performed by using the field of values of the preconditioned linear system. As an example, a saddle point problem obtained from a mixed finite element discretization of the Oseen equations is considered. The convergence estimates obtained by using a field–of–values analysis are independent of the discretization parameter h. Several computational experiments supplement the theoretical results and illustrate the performance of the method. Received March 20, 1997 / Revised version received January 14, 1998     

Error estimate for convection problem with characteristics method

We consider the numerical approximation of a first order steady-state convective problem by the method of characteristics with pseudo-time step k and P _r discontinuous finite elements on a mesh T _h. We show, when the entry flow field is Ω filling and under a technical hypothesis, the existence and the uniqueness for the continuous and the discrete variational problems (P^k) and P( _h ^k ) and we give an error estimate. This revised version was published online in June 2006 with corrections to the Cover Date.     

Gradient based enhanced finite element formulation for diffuse phase interfaces

Interfaces between adjacent phases, so-called domain walls, appear as non-linear gradients of order parameters in diffuse phase field models. Usually, the interface width is much smaller than the dimension of the simulated region. Since the position of domain walls is not known a priori the maximum size of finite elements needs to be adapted to the length scale of interfaces within the entire region. We suggested a selective finite element method to improve the numerical solution of diffuse phase field models [1, 2]. It enhances the finite element interpolation space using supplementary local degrees of freedom. However, corresponding additional nodes are strictly located in the interior of elements, thus, C⁰-continuity at element border is guaranteed. Since C⁰-continuity limits the performance of this method we propose in this paper a relaxation of C⁰-requirements perpendicular to the gradient of the order parameter. Therefore, the direction of interfaces is analyzed as additional information for further adaptive improvement of the interpolation space. A dual phase field model is used to validate the proposed method. The analytical solution of a stationary domain wall allows error analysis of regular and distorted finite element meshes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Exact Results in the Finite Element Method

We prove that the finite element method for one-dimensional problems yields no discretization error at nodal points provided the shape functions are appropriately chosen. Then we consider a biharmonic problem with mixed boundary conditions and the weak solution u. We show that the Galerkin approximation of u based on the so-called biharmonic finite elements is independent of the values of u in the interior of any subelement.     

The Mortar Element Method with Locally Nonconforming Elements

We consider a discretization of linear elliptic boundary value problems in 2-D by the new version of the mortar finite element method which uses locally nonconforming Crouzeix-Raviart elements. We show that if a solution of the original differential problem belongs to the space H ²(), then an error is of the same order as in the standard nonconforming finite element method. We also propose an additive Schwarz method of solving the discrete problem and show that its rate of convergence is almost optimal.     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.     

A unified approach to a posteriori error estimation using element residual methods

Summary This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283–301 (1985)].     

A compact C0 discontinuous Galerkin method for Kirchhoff plates

A compact C⁰ discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C⁰ finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C⁰ interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H¹‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015     

Adini‐Q1‐P3 FEM for general elastic multi‐structure problems

An Adini‐Q₁‐P₃ finite element method is introduced to solve general elastic multi‐structure problems, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized by conforming linear (bilinear or trilinear) elements, and transverse displacements on plates and rods are discretized by Adini elements and Hermite elements of third order, respectively. The unique solvability and optimal error estimates in the energy norm are established for the discrete method, whose numerical performance is illustrated by some numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1092–1112, 2011