Adaptive multivlevel methods in three space dimensions

Folkmar Bornemann, Bodo Erdmann, Ralf Kornhuber

Research output: Contribution to journalArticlepeer-review

99 Scopus citations


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.

Original languageEnglish
Pages (from-to)3187-3203
Number of pages17
JournalInternational Journal for Numerical Methods in Engineering
Issue number18
StatePublished - 30 Sep 1993
Externally publishedYes


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