An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

Dörfler, W.; Rumpf, M.

doi:10.1090/S0025-5718-98-00993-4

An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation
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by W. Dörfler and M. Rumpf PDF

Math. Comp. 67 (1998), 1361-1382 Request permission

Abstract:

We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape. A posteriori error estimates are given in the energy norm and the $L^2$-norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

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