17th GAMM-Seminar Leipzig on
Construction of Grid Generation Algorithms

Max-Planck-Institute for Mathematics in the Sciences
Inselstr. 22-26, D-04103 [O->]Leipzig
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  17th GAMM-Seminar
February, 1st-3rd, 2001
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  Abstract O. Shishkina, Thu, 14.30-14.55 Previous Contents Next  
  On Solving Elliptic PDE's via Adaptive Mesh Refinement
O. Shishkina (State University, Moscow)

An adaptive mesh refinement procedure for solving elliptic problems in polygonal domains is presented. Each step of this procedure deals with a fixed triangulation. The mesh refinement procedure uses an a posteriori error indicator (EI), which estimates an expected variation of the finite element solution in any point of triangulation while complementing the standard nodal basis by a probe piecewise linear function associated with this point. For each edge of the concerned triangulation the "edge point" is found, which is supposed to be close enough to the point on the edge, where the value of EI has maximum. (A point, where an interpolating polynomial for EI of the fourth degree has maximum, is taken as the edge point). At the first steps of mesh refinement procedure (when the mesh is still coarse) all the edge points complement the set of vertices in the triangulation. Then all the edge points in any triangle are connected with each other and the finite element problem is solved in the set of piecewise linear functions associated with the complemented set of vertices. At the last steps only those edge points, where the value of EI is large enough, complement the set of vertices. In this paper some numerical experiments are presented to compare some versions of our approach with the standard finite element method. All the problems for which the EI is exact are also characterized.

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