Concepts of an adaptive hierarchical finite element code

Abstract

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.

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

“…In the case of standard conforming finite element discretizations, the hierarchical basis error estimator has been investigated by Deuflhard, Leinen, Yserentant [14]. An excellent overview is given by Bornemann et al [8].…”

“…In Sections 4 and 5, we present two types of hierarchical error estimators that are strongly related and require an adequate saturation assumption (cf., e.g., [6,14,15,21,25,27]). In particular, for the derivation of the first hierarchical error estimator, which is dealt with in Section 4, we start from an approximation of the defect problem in a higher order ansatz space followed by a localization in terms of an appropriate hierarchical two-level splitting.…”

A multilevel successive iteration method for nonlinear elliptic problems

“…In contrast with FMG, Deuflhard [9] proposed a cascadic iteration algorithm which employs nested iterations using the conjugate gradient (CG) method or the preconditioned conjugate gradient (PCG) method instead of using MG at each level. Some adaptive strategies are also proposed, see, e.g., [9,10]. The efficiency of the cascadic algorithm has been demonstrated numerically in [9,10], and the comparison is made with the numerical results obtained by using the multilevel preconditioner of Bramble, Pasciak and Xu [5] and the hierarchical preconditioner of Yserentant [27].…”

“…It is also efficient for a large class of problems, as demonstrated in [4,9,10]. Let the index of the final level be J and that of the current level be j.…”

“…Some adaptive strategies are also proposed, see, e.g., [9,10]. The efficiency of the cascadic algorithm has been demonstrated numerically in [9,10], and the comparison is made with the numerical results obtained by using the multilevel preconditioner of Bramble, Pasciak and Xu [5] and the hierarchical preconditioner of Yserentant [27]. The main feature of the cascadic iteration is coarse-grid-correction free, and as a result it can be viewed as a one-way multigrid method.…”