
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.

