Abstract This paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2D elliptic problems on general domains. The starting point for this new development is the recent work on hierarchical finite element bases by H. Yserentant (Numer. Math.49, 379–412 (1986)). It is shown that this approach permits a flexible balance among iterative solver, local error estimator, and local mesh refinement device—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.
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