Adaptive multivlevel methods in three space dimensions

We consider the approximate solution of self-adjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non-uniform tetrahedral mesh which is generated adaptively. The arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner. Here, the accuracy of the iterative solution is coupled with the discretization error. As the performance of hierarchical bases preconditioners deteriorates in three space dimensions, the BPX preconditioner is used, taking special care of an efficient implementation. Reliable a posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements. To illustrate the theoretical results, we consider a familiar model problem involving reentrant corners and a real-life problem arising from hyperthermia, a recent clinical method for cancer therapy.

[1]  F. B. Ellerby,et al.  Numerical solutions of partial differential equations by the finite element method , by C. Johnson. Pp 278. £40 (hardback), £15 (paperback). 1988. ISBN 0-521-34514-6, 34758-0 (Cambridge University Press) , 1989, The Mathematical Gazette.

[2]  I. Babuška,et al.  Direct and inverse error estimates for finite elements with mesh refinements , 1979 .

[3]  W. Dahmen,et al.  Multilevel preconditioning , 1992 .

[4]  M. E. Go Ong,et al.  Hierachical basis preconditioners for second order elliptic problems in three dimensions , 1989 .

[5]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[6]  R. Hoppe,et al.  Adaptive multilevel methods for obstacle problems , 1994 .

[7]  Harry Yserentant,et al.  A basic norm equivalence for the theory of multilevel methods , 1993 .

[8]  Folkmar A. Bornemann,et al.  An adaptive multilevel approach to parabolic equations : II. Variable-order time discretization based on a multiplicative error correction , 1991, IMPACT Comput. Sci. Eng..

[9]  Folkmar Bornemann,et al.  KASKADE - Numerical Experiments. , 1991 .

[10]  Hans D. Mittelmann,et al.  PLTMG: A Software Package for Solving Elliptic Partial Differential Equations (Randolph E. Bank) , 1994, SIAM Review.

[11]  J. Pasciak,et al.  Parallel multilevel preconditioners , 1990 .

[12]  Eberhard Bänsch,et al.  Local mesh refinement in 2 and 3 dimensions , 1991, IMPACT Comput. Sci. Eng..

[13]  Harry Yserentant,et al.  Two preconditioners based on the multi-level splitting of finite element spaces , 1990 .

[14]  H. Yserentant.  Erratum. On the Multi-Level Splitting of Finite Element Spaces.(Numer. Math. 49, 379-412 (1986)). , 1986 .

[15]  Philippe G. Ciarlet,et al.  The Finite Element Method for Elliptic Problems. , 1981 .

[16]  P. Deuflhard,et al.  Numerical approaches to treatment planning in deep RF-hyperthermia. , 1989, Strahlentherapie und Onkologie : Organ der Deutschen Rontgengesellschaft ... [et al].

[17]  Michael Griebel,et al.  Multilevel Algorithms Considered as Iterative Methods on Semidefinite Systems , 1994, SIAM J. Sci. Comput..

[18]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[19]  Peter Deuflhard,et al.  Concepts of an adaptive hierarchical finite element code , 1989, IMPACT Comput. Sci. Eng..

[20]  I. Babuska,et al.  Finite Element Analysis , 2021 .