A Parallel Solver for a Preconditioned Space-Time Boundary Element Method for the Heat Equation

Semi-analytic integration for a parallel space-time boundary element method modeling the heat equation

“…We have described details of the applied Galerkin methods in Section 3. In addition we use the matrix V 11 h as the essential part of an operator preconditioner for D h [25,26,7], where V 11 h is the realisation of the single-layer operator for functions piecewise linear and globally continuous in space. We check the convergence of the approximations u h and w h corresponding to u and w to the known Cauchy data on a sequence of uniformly refined meshes Σ h .…”

“…We aim to provide a complete and (hopefully) error-free presentation of details on the implementation of a Galerkin boundary element method for the three-dimensional heat equation considering all boundary integral operators. Galerkin methods have been considered in, e.g., [2,3,6,7] for 2d and [8,9,10] for 3d. Typically, implementational aspects are discussed only briefly and a lot of effort is necessary to transform the theoretical results into a performant computer code.…”

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

“…precluding point singularities admitted in the present work. The work [12] focuses again on space-time discretization of the heat equation, and addresses the solver complexity. A space-time DG discretization for the parabolic Navier-Stokes equation was proposed in [41].…”

“…where P p (F x ) denotes the space of polynomials of degree at most p ∈ N 0 on F x , and the constants C 2|1 , C ∞|2 depend on p. The inequalities in (11) are obvious, while the inverse inequalities in (12) follow from the equivalence of norms in finite dimensional spaces and scaling; note that C ∞|2 ≥ 1. The constants in (12) depend linearly on the polynomial degree. Denote by P p the Legendre polynomials and by…”

“…The inverse inequalities (12) involve spaces of single-variable polynomials defined on segments; in the error analysis carried out in the next sections, we do not use inverse inequalities for spaces defined on polygons or polyhedra, so the shapes of the mesh elements do not play any role in this respect.…”