Concepts of an adaptive hierarchical finite element code

The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2-D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem.