An optimally convergent adaptive mixed finite element method

We prove convergence and optimal complexity of an adaptive mixed finite element algorithm, based on the lowest-order Raviart–Thomas finite element space. In each step of the algorithm, the local refinement is either performed using simple edge residuals or a data oscillation term, depending on an adaptive marking strategy. The inexact solution of the discrete system is controlled by an adaptive stopping criterion related to the estimator.     

Error estimates for a mixed finite volume method for the p-Laplacian problem

In this work we propose and analyze a mixed finite volume method for the p-Laplacian problem which is based on the lowest order Raviart–Thomas element for the vector variable and the P1 nonconforming element for the scalar variable. It is shown that this method can be reduced to a P1 nonconforming finite element method for the scalar variable only. One can then recover the vector approximation from the computed scalar approximation in a virtually cost-free manner. Optimal a priori error estimates are proved for both approximations by the quasi-norm techniques. We also derive an implicit error estimator of Bank–Weiser type which is based on the local Neumann problems.This work was supported by the Post-doctoral Fellowship Program of Korea Science & Engineering Foundation (KOSEF).     

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.     

Fortin operator and discrete compactness for edge elements

Summary. The basic properties of the edge elements are proven in the original papers by Nédélec [22,23] In the two-dimensional case the edge elements are isomorphic to the face elements (the well-known Raviart–Thomas elements [24]), so that all known results concerning face elements can be easily formulated for edge elements. In three-dimensional domains this is not the case. The aim of the present paper is to show how to construct a Fortin operator which converges uniformly to the identity in the spirit of [5,4]. The construction is given for any order tetrahedral edge elements in general geometries. We relate this result to the well-known commuting diagram property and apply it to improve the error estimate for a mixed problem which involves edge elements. Finally we show that this result can be applied to the analysis of the approximation of the time-harmonic Maxwell's system. Received March 22, 1999 / Revised version received September 23, 1999 / Published online July 12, 2000     

Solution of Maxwell equation in axisymmetric geometry by Fourier series decompostion and by use of H (rot) conforming finite element

Summary. This study deals with the mathematical and numerical solution of time-harmonic Maxwell equation in axisymmetric geometry. Using Fourier decomposition, we define weighted Sobolev spaces of solution and we prove expected regularity results. A practical contribution of this paper is the construction of a class of finite element conforming with the H (rot) space equipped with the weighted measure rdrdz. It appears as an extension of the well-known cartesian mixed finite element of Raviart-Thomas-Nédélec [11]–[15]. These elements are built from classical lagrangian and mixed finite element, therefore no special approximations functions are needed. Finally, following works of Mercier and Raugel [10], we perform an interpolation error estimate for the simplest proposed element. Received March 15, 1996 / Revised version received November 30, 1998 / Published online December 6, 1999     

Auxiliary space preconditioning in H 0(curl; Ω)

We adapt the principle of auxiliary space preconditioning as presented in [J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215–235.] to H (curl; ω)-elliptic variational problems discretized by means of edge elements. The focus is on theoretical analysis within the abstract framework of subspace correction. Employing a Helmholtz-type splitting of edge element vector fields we can establish asymptotic h-uniform optimality of the preconditioner defined by our auxiliary space method. This author was fully supported by Hong Kong RGC grant (Project No. 403403) This author acknowledges the support from a Direct Grant of CUHK during his visit at The Chinese University of Hong Kong.     

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

First-order convergence of multi-point flux approximation on triangular grids and comparison with mixed finite element methods

In this paper we show first-order convergence of a multi-point flux approximation control volume method (MPFA) on unstructured triangular grids. In this approach the flux approximation is derived directly in the physical space. In order to do this, we introduce a perturbed mixed finite element method that is equivalent to the MPFA scheme and prove the first-order convergence of this approach. Moreover, we carefully compare the computational performance properties of the MPFA method with those of a lowest order Raviart–Thomas and Brezzi–Douglas–Marini mixed finite element approximation.     

Symmetric coupling of boundary elements and Raviart–Thomas-type mixed finite elements in elastostatics

Summary. Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical examples are included. Received February 21, 1995 / Revised version received December 21, 1995     

A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.This work was supported by the GdR MoMaS, CNRS-2439, ANDRA, BRGM, CEA, EdF, France.     

Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension

We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. In particular we define regularization operators which, combined with the standard interpolators, enable us to prove discrete Poincaré–Friedrichs inequalities and discrete Rellich compactness for finite element spaces of differential forms of arbitrary degree on compact manifolds of arbitrary dimension.     

On the adaptive coupling of FEM and BEM in 2–d–elasticity

Summary. This paper concerns the combination of the finite element method (FEM) and the boundary element method (BEM) using the symmetric coupling. As a model problem in two dimensions we consider the Hencky material (a certain nonlinear elastic material) in a bounded domain with Navier–Lamé differential equation in the unbounded complementary domain. Using some boundary integral operators the problem is rewritten such that the Galerkin procedure leads to a FEM/BEM coupling and quasi–optimally convergent discrete solutions. Beside this a priori information we derive an a posteriori error estimate which allows (up to a constant factor) the error control in the energy norm. Since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh–refinement we state adaptive algorithms for the –version of the FEM/BEM–coupling. Illustrating numerical results are included. Received April 15, 1994 / Revised version received January 8, 1996     

Development and Analysis of Higher Order Finite Volume Methods over Rectangles for Elliptic Equations

Currently used finite volume methods are essentially low order methods. In this paper, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. In particular, we restrict ourselves to finite volume methods posed over rectangular partitions and begin by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of index two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we derive finite volume methods of all orders from the Raviart–Thomas two-dimensional rectangular elements; we also find finite volume methods to associate with BDFM ₂ three-dimensional rectangles. In each case, we obtain optimal error estimates for both the scalar variable and the recovered flux.     

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.     

On a hybrid boundary element method

Summary. In this paper we study a symmetric boundary element method based on a hybrid discretization of the Steklov–Poincaré operator well suited for a symmetric coupling of finite and boundary elements. The representation used involves only single and double layer potentials and does not require the discretization of the hypersingular integral operator as in the symmetric formulation. The stability of the hybrid Galerkin discretization is based on a BBL–like stability condition for the trial spaces. Numerical examples confirm the theoretical results. Received December 15, 1997 / Revised version received December 21, 1998/ Published online November 17, 1999     

Multigrid in H (div) and H (curl)

Summary. We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces and in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products. Received June 12, 1998 / Revised version received March 12, 1999 / Published online January 27, 2000     

Schwarz methods for inequalities with contraction operators

We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier–Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type.     

An adaptive stabilized finite element method for the generalized Stokes problem

In this work we present an adaptive strategy (based on an a posteriori error estimator) for a stabilized finite element method for the Stokes problem, with and without a reaction term. The hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble-like functions. An equivalence result between the norm of the finite element error and the estimator is given, where the dependence of the constants on the physics of the problem is explicited. Several numerical results confirming both the theoretical results and the good performance of the estimator are given.     

Superconvergence for rectangular mixed finite elements

Summary In this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems. For the rectangular finite elements of Raviart and Thomas [19] and for those of Brezzi et al. [4] we prove that the distance inL ² between the approximate solution and a projection of the exact one is of higher order than the error itself.This result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing.     

Finite element approximation of elliptic control problems with constraints on the gradient

We consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state. We present a tailored finite element approximation to this optimal control problem, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order Raviart–Thomas mixed finite element. Pointwise bounds on the gradient variable are enforced in the elements of the triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and three space dimensions. A numerical example confirms our analytical findings.