Adaptive error control for multigrid finite element - Springer Link

Computing55, 271-288(1995)

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9 Springer-Verlag 1995 Printed in Austria

Adaptive Error Control for Multigrid Finite E l e m e n t Methods R. Becker, Heidelberg, C. Johnson, G6teborg, and R. Rannacher, Heidelberg Received October 24, 1994; revised June 6, 1995

Abstract - - Zusammenfassung Adaptive Error Control for Multigrid Finite Element Methods. We consider the problem of adaptive error control in the finite element method including the error resulting from inexact solution of the discrete equations. We prove a posteriori error estimates for a prototype elliptic model problem discretized by the finite element method with a canonical multigrid algorithm. The proofs are based on a combination of so-called strong stability and the orthogonality inherent in both the finite element method and the multigfid algorithm.

AMS Subject Classification: 65F10

Key words: Finite elements, multigrid methods, error control. Adaptive Fehlerkontrolle fiir Finite-Elemente-Mehrgitter-Methoden. Wir behandeln das Problem einer adapfiven Fehlerkontrolle bei Finite-Elemente-Methoden unter Einschlug des Fehlers, der durch ungenaue L6sung der diskreten Gleichungen entsteht. Wir beweisen A-posterioriFehlerabschgtzungen fur ein elliptisches Modellproblem, welches mit linearen finiten Elementen diskretisiert wird. Die diskreten Gleichungen werden mit Hilfe des kanonischen Finite-ElementeMehrgitterverfahrens gel6st. Die Beweise beruhen auf der Kombination der "starken" Stabilitgtseigenschaft des zugrundeliegenden Differentialoperators und der Galerkin-Orthogonalitfit sowohl des Finite-Elemente- als auch des Mehrgitterverfahrens.

1. Introduction In this note we consider the problem of reliable and efficient adaptive error control in the finite element method with particular emphasis on the error resulting from solving the discrete equations approximately using iterative methods. The discrete equations may be solved exactly (neglecting round-off) by direct Gaussian elimination for problems of small or medium size, but for large size problems, in particular in three dimensions, direct methods are too work-intensive and only iterative methods such as multigrid methods or preconditioned conjugate gradient type methods may be used. There is an extensive literature of a priori type on the convergence properties of such iterative methods, in particular concerning linear elliptic problems, but objective stopping criteria for the iterations of a posteriori type seem to be lacking. Usually, ad hoc stopping criteria are used, e.g. requiring an initial (algebraic) residual to be reduced by a

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certain ad hoc factor, but these criteria have no clear connection to the actual error in the corresponding approximate solution, which is the quantity of interest. This leaves the user of iterative solutions methods in a serious dilemma: With no objective stopping criterion available, one has either to continue the iterations until the discrete solution error is practically "zero", which increases the computational cost with possibly no gain in the overall precision, or take the risk of stopping the iterations prematurely. In the first case there would be a loss of efficiency and in the second a loss of reliability. It would seem natural to stop the iterations when the error from the approximate solution of the discrete equations is comparable to the error from the finite element discretization itself, but objective stopping criteria giving such a balance are not available in the literature. It is possible that this has prevented the use of iterative methods by many practitioners. Thus finding objective stopping criteria for iterative solution techniques for finite element equations appears to be an important open problem. A solution to this problem can only be obtained by combining aspects of the underlying partial differential equations and the corresponding finite element discretization with aspects of the iterative discrete solution algorithm. A "pure" numerical linear algebra point of view, for instance based on the condition number of the stiffness matrix, does not appear to be able to lead to a balance of discretization and solution errors. In the series of papers [3]-[6] and the forthcoming book [2] a general theory for automatic adaptive control of the discretization error in the finite element method is developed. The objective is to design and analyze algorithms for reliable and efficient automatic control of the error in a norm and on a tolerance level given by the user. For instance, algorithms are designed for adaptive control of the error in the maximum norm for the stresses or displacements in linear elasticity. The adaptive algorithms are based on a posteriori error estimates involving estimates of the exact residual of the finite element solution, where the exact residual is obtained inserting the finite element solution into the given differential equation. The proofs of the a posteriori error estimates are based on a combination of (i) strong stability (of an associated continuous dual problem) and (ii) the Galerkin orthogonality built in the finite element method. In the derivation of these estimates, the discrete equations are supposed to be solved exactly, which in particular guarantees "full" Galerkin orthogonality. In this note we now consider in a model case a situation where the discrete equations are solved only approximately, in which case the total error will have contributions from two sources: (a) the discretization error in the finite element method, (b) the discrete solution error due to inexact solution of the discrete equations. Our goal is to develop algorithms for reliable and efficient automatic control in a variety of norms of both the discretization error and the solution error. In particular we seek appropriate stopping criteria for iterative methods for the approximate solution of the discrete equations, such that both the discretization error and the discrete solution error will be controlled on a given tolerance level in a given norm in an efficient and reliable way. In the present paper we carry out this program in a simple model case

Adaptive Error Control for Multigrid Finite Element Methods

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considering a standard finite element method for the Poisson equation solving the discrete equations with a multigrid algorithm, and with error control in the L 2 or H I norms. We emphasize the prototype character of this paper seeking to exhibit an essential feature of the problem of automatic error control of the "complete" discretization process including both finite element discretization and discrete solution errors. The considerations of this note seem to be applicable to a large variety of problems and finite element discretizations when coupled with the material of [3]-[6]. We hope to give more details in future work. Note that there are also other sources of errors resulting, e.g., from quadrature and round-off which we do not consider here, which also in principle should be controllable through a posteriori error estimates. We hope to return to the topic of automatic control of quadrature and round-off errors in future work. We shall consider the standard finite element method based on piecewise linear approximation with a multigrid solver applied to a standard elliptic model problem, the Poisson's equation on a polygonal domain. We prove a posteriori error estimates in the H a and L 2 norms for the discretization and solution errors, design corresponding adaptive algorithms and discuss their reliability and efficiency. Finally, we present some numerical results for supporting the theory.

2. Poisson's Equation Consider the Poisson's equation with homogeneous Dirichlet boundary conditions: Find u = u(x) such that - A u = f in~2, u =0 on00, (1) where O is a bounded domain in R 2 with boundary 3g~, x = (xt, x2) , A is the Laplace operator, and f = f ( x ) is given data. The standard variational form of (1) reads: Find u ~ H~(IJ) such that

(Vu, V u ) = ( f , v ) Vv ~ H ~ ( a ) , (2) where (w, v) = fa wv dx, (Vw, 17v) = fa Vw. Vv dx, and H~(~2) as usual denotes the completion of the space of test functions with respect to the norm of the Sobolev space H i ( O ) . Further, we use the notation [['1] for the norm of the Lebesgue space LZ(g~) over O. We recall that by Poincarg's inequality, HV"I] is a norm in H01 ( O ) equivalent to the Hl(O)-norm. The existence of a unique solution of (2) follows by the Ritz representation theorem if f ~ H - 1 (O), where H -1 ( O ) is the dual space of H01(~2) with norm IlfllH-i(m =

sup

( f , v).

v ~ H0~(O),lt7 vii= 1

We recall basic strong stability estimates for the solution of (2) to be used in the proofs of the a posteriori error estimates below. The estimates are referred to as "strong", because derivatives of the solution u are estimated.

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Lemma 1. The solution u o f (2) satisfies

IlVull _< CsllfllH-l