Joachim Schöberl: Hierarchical shape functions based on explicit polynomial extension operators
The goal of this work is the construction of fast iterative solvers for matrix equations
arising from high order finite elements. We focus on cheap block-Jacobi and
block-Gauss-Seidel iterations, where the blocks are defined by the shape functions
associated to edge-, face- and interior-nodes. Of course, the speed of convergence
depends on the choice of the shape functions.
We present new shape functions leading to nearly optimal iteration numbers. The
construction is based on polynomial extension operators. By the help of symbolic
computing we could derive cheap recursion formulas for the efficient computation
of the shape functions.
Numerical results for 2D and 3D problems are presented.