$hp$-version additive Schwarz algorithms on triangular meshes

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. In relation to the definition of these coordinate functions within the elements, we also distinguish two cases: arbitrary and so-called discrete quasiharmonic coordinate functions. The latter are obtained by means of explicitely given and nonexpensive prolongation operators. In all these situations, we are able to suggest preconditioners which are spectrally equivalent or almost spectrally equivalent to the global stiffness matrix, which require only element-by-element and edge-by-edge operations, thus, providing a high level of parallelization. They also considerably reduce the computational cost. In our domain decomposition algorithms, we essentially use prolongation operators in the polynomial spaces from the interface boundary inside subdomains of the decomposition following the approach initially used by S. Ivanov and V. Korneev for the $hp$-version with quadrilateral elements.