Additive Schwarz algorithms for solving hp-version finite element systems on triangular meshes Original Research Article

Highly parallelizable domain decomposition Dirichlet–Dirichlet solvers for hp-version finite element methods on angular quasiuniform triangular meshes are studied under different assumptions on a reference element. The edge coordinate functions of a reference element are allowed to be either nodal with special choices of nodes, or hierarchical polynomials of several types. These coordinate functions are defined within elements as being arbitrary or discrete quasi-harmonic coordinate functions. The latter are obtained from explicit and inexpensive prolongation operators. In all situations, we are able to suggest preconditioners which are spectrally equivalent to the global stiffness matrix, which only require element-by-element and edge-by-edge operations, and which reduce computational cost. In this way, elimination is avoided when dealing with the interface problem. The domain decomposition algorithms essentially use prolongation operators from the interface boundary inside the subdomains of the decomposition according to the approach initially used for the hp-version finite element methods with quadrilateral elements [S.A. Ivanov, V.G. Korneev, Izv. Vyssh. Uchebn. Zaved. 395 (1995) 62–81; Technische Universität Chemnitz–Zwickau, Preprint SPC 95-35, 1995, 1–15, and Preprint SPC 95-36, 1995, 1–14; Math. Modeling 8 (1996) 63–73].