Ansatz Spaces Research Articles

An iterative substructuring method for the $p$ -version of the boundary element method for hypersingular integral operators in three dimensions

We study preconditioners for the \(p\)-version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioners are based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis functions. We consider a so-called wire basket preconditioner and a non-overlapping additive Schwarz method based on the complete natural splitting, i.e. with respect to the nodal, edge and interior functions, as well as an almost diagonal preconditioner. In any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the mesh size. For all these methods we prove that the resulting condition numbers are bounded by \(C(1+\log p)\). Here, \(p\) is the polynomial degree of the ansatz functions and \(C\) is a constant which is independent of \(p\) and the mesh size of the underlying boundary element mesh. Numerical experiments supporting these results are reported.

Multilevel iterative solution and adaptive mesh refinement for mixed finite element discretizations

We consider the numerical solution of elliptic boundary value problems by mixed finite element discretizations on simplicial triangulations. Emphasis is on the efficient iterative solution of the discretized problems by multilevel techniques and on adaptive grid refinement. The iterative process relies on a preconditioned conjugate gradient iteration in a suitably chosen subspace with a multilevel preconditioner of hierarchical type that can be constructed by means of appropriate multilevel decompositions of the mixed ansatz spaces. Using the Dryja-Widlund theory of additive Schwarz iterations, we show that the spectral condition number of the preconditioned stiffness matrix asymptotically exhibits a quadratic growth in the refinement level as it is the case in the standard conforming approach. The adaptive grid refinement is based on an efficient and reliable a posteriori error estimator for the total error in the flux which can be established by a defect correction in higher order mixed ansatz spaces combined with a hierarchical two-level splitting.

Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods

We consider mixed finite element discretizations of second order elliptic boundary value problems. Emphasis is on the efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations. In particular, we present two multilevel solvers, the first one relying on ideas from domain decomposition and the second one resulting from mixed hybridization. Local refinement of the underlying triangulations is done by efficient and reliable a posteriori error estimators which can be derived by a defect correction in higher order ansatz spaces or by taking advantage of superconvergence results. The performance of the algorithms is illustrated by several numerical examples.