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Fast adaptive domain decomposition algorithms for \(hp\)-discretizations of 2-\(D\) and 3-\(D\) elliptic equations: recent advances. (English) Zbl 1069.65136

Summary: A fast solver for \(hp\)-adaptive computations must possess definite properties. In particular, it should be efficient for discretizations in which for each finite element, each face and each edge the orders of the subspaces of the internal polynomials (on the respective subset of the reference element) may be different. An efficient domain decomposition (DD) preconditioner for the hierarchical \(hp\) discretizations of 3D elliptic equations of second-order, possessing the pointed out properties, is presented. It is based on the fast inexact solvers for the internal Dirichlet problems on finite elements and faces and wire basket preconditioning. We also justify the use of inexact solvers for prolongations inside elements from the interelement boundary and from the wire basket on faces. The DD preconditioner is almost optimal with respect to \(p\) in the total arithmetical cost and provides a high level of parallelization.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65Y05 Parallel numerical computation
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