Convergence and optimal complexity of adaptive finite element eigenvalue computations

In this paper, an adaptive finite element method for elliptic eigenvalue problems is studied. Both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved. The analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper. This work was partially supported by the National Science Foundation of China under grant 10425105 and the National Basic Research Program under grant 2005CB321704.     

On the Galerkin finite element approximations to multi-dimensional differential and integro-differential parabolic equations

The maximum norm error estimates of the Galerkin finite element approximations to the solutions of differential and integro-differential multi-dimensional parabolic problems are considered. Our method is based on the use of the discrete version of the elliptic-Sobolev inequality and some operator representations of the finite element solutions. The results of the present paper lead to the error estimates of optimal or almost optimal order for the case of simplicial Lagrangian piecewise polynomial elements.     

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.     

Two-grid finite volume element method for linear and nonlinear elliptic problems

Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates. The work is supported by the National Natural Science Foundation of China (Grant No: 10601045).     

Iterative schemes for mixed finite element methods with applications to elasticity and compressible flow problems

Summary Iterative schemes for mixed finite element methods are proposed and analyzed in two abstract formulations. The first one has applications to elliptic equations and incompressible fluid flow problems, while the second has applications to linear elasticity and compressible Stokes problems. These schemes are constructed through iteratively penalizing the mixed finite element scheme, of which iterated penalty method and augmented Lagrangian method are special cases. Convergence theorems are demonstrated in abstract formulations in Hilbert spaces, and applications to individual physical problems are considered as examples. Theoretical analysis and computational experiments both show that the proposed schemes have very fast convergence; a few iterations are normally enough to reduce the iterative error to a prescribed precision. Numerical examples with continuous and discontinuous coefficients are presented.     

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.     

A defect-correction method for unsteady conduction-convection problems II: Time discretization

In this paper, a fully discrete defect-correction mixed finite element method (MFEM) for solving the non-stationary conduction-convection problems in two dimension, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect-correction technique. The stability and the error analysis are derived. The theory analysis shows that our method is stable and has a good convergence property. Some numerical results are also given, which show that this method is highly efficient for the unsteady conduction-convection problems.     

A characteristic finite element method for optimal control problems governed by convection-diffusion equations

In this paper we analyze a characteristic finite element approximation of convex optimal control problems governed by linear convection-dominated diffusion equations with pointwise inequality constraints on the control variable, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by either piecewise constant functions or piecewise linear discontinuous functions. A priori error estimates are derived for the state, co-state and the control. Numerical examples are given to show the efficiency of the characteristic finite element method.     

Nonconforming finite elements and the cascadic multi-grid method

Summary. We derive sufficient conditions under which the cascadic multi-grid method applied to nonconforming finite element discretizations yields an optimal solver. Key ingredients are optimal error estimates of such discretizations, which we therefore study in detail. We derive a new, efficient modified Morley finite element method. Optimal cascadic multi-grid methods are obtained for problems of second, and using a new smoother, of fourth order as well as for the Stokes problem. Received February 12, 1998 / Revised version received January 9, 2001 / Published online September 19, 2001     

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

Summary Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP ^th -degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 90-0194; by the U.S. Army Research Office under Contract Number DAAL03-91-G-0215; and by the National Science Foundation under Institutional Infrastructure Grant Number CDA-8805910     

Error estimates of the lumped mass finite element method for semilinear elliptic problems

We are concerned with the semilinear elliptic problems. We first investigate the L²-error estimate for the lumped mass finite element method. We then use the cascadic multigrid method to solve the corresponding discrete problem. On the basis of the finite element error estimates, we prove the optimality of the proposed multigrid method. We also report some numerical results to support the theory.     

On superconvergence techniques

A brief survey with a bibliography of superconvergence phenomena in finding a numerical solution of differential and integral equations is presented. A particular emphasis is laid on superconvergent schemes for elliptic problems in the plane employing the finite element method.     

An algorithm using the finite volume element method and its splitting extrapolation

This paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation. This method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation, such as a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than a Richardson extrapolation. Because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition.     

Two-level hierarchically preconditioned conjugate gradient methods for solving linear elasticity finite element equations

In this paper we study and compare some preconditioned conjugate gradient methods for solving large-scale higher-order finite element schemes approximating two- and three-dimensional linear elasticity boundary value problems. The preconditioners discussed in this paper are derived from hierarchical splitting of the finite element space first proposed by O. Axelsson and I. Gustafsson. We especially focus our attention to the implicit construction of preconditioning operators by means of some fixpoint iteration process including multigrid techniques. Many numerical experiments confirm the efficiency of these preconditioners in comparison with classical direct methods most frequently used in practice up to now.     

Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations

In this paper, we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by Stokes equations. We obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods, which are the patch recovery technique and the least-squares surface fitting method. Our results are based on some regularity assumption for the Stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions.     

Error estimators for advection–reaction–diffusion equations based on the solution of local problems

This paper deals with a posteriori error estimates for advection–reaction–diffusion equations. In particular, error estimators based on the solution of local problems are derived for a stabilized finite element method. These estimators are proved to be equivalent to the error, with equivalence constants eventually depending on the physical parameters. Numerical experiments illustrating the performance of this approach are reported.     

Error estimates of mixed finite element methods for quadratic optimal control problems

In this paper, we investigate the error estimates for the solutions of optimal control problems by mixed finite element methods. The state and costate are approximated by Raviart-Thomas mixed finite element spaces of order k and the control is approximated by piecewise polynomials of order k. Under the special constraint set, we will show that the control variable can be smooth in the whole domain. We derive error estimates of optimal order both for the state variables and the control variable.     

A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

Interior estimates for nonconforming finite element methods

Summary. Interior error estimates are derived for a wide class of nonconforming finite element methods for second order scalar elliptic boundary value problems. It is shown that the error in an interior domain can be estimated by three terms: the first one measures the local approximability of the finite element space to the exact solution, the second one measures the degree of continuity of the finite element space (the consistency error), and the last one expresses the global effect through the error in an arbitrarily weak Sobolev norm over a slightly larger domain. As an application, interior superconvergences of some difference quotients of the finite element solution are obtained for the derivatives of the exact solution when the mesh satisfies some translation invariant condition. Received December 29, 1994     

Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems,part I: Algorithms and numerical results

Summary We describe sequential and parallel algorithms based on the Schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the Raviart-Thomas finite element spaces. These lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block Gauss-Seidel or block Jacobi methods with overlapping blocks. The indefiniteness requires special treatment. The sub-blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of Dryja and Widlund for standard elliptic problems. If there is sufficient overlap between the subdomains, the algorithm converges with a rate independent of the mesh size, the number of subdomains and discontinuities of the coefficients. Extensions of the above algorithms to the case of local grid refinement is also described. Convergence theory for these algorithms will be presented in a subsequent paper.This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003, while the author was a graduate student at New York University, and in part by the Army Research Office under Grant DAAL 03-91-G-0150, while the author was a Visiting Assistant Researcher at UCLA