A DG Space-Time Domain Decomposition Method

In this paper we present a hybrid domain decomposition approach for the parallel solution of linear systems arising from a discontinuous Galerkin (DG) finite element approximation of initial boundary value problems. This approach allows a general decomposition of the space–time cylinder into finite elements, and is therefore applicable for adaptive refinements in space and time. 1 A Space–Time DG Finite Element Method As a model problem we consider the transient heat equation ∂tu(x, t)−∆u(x, t) = f(x, t) for (x, t) ∈ Q := Ω × (0, T ), (1) u(x, t) = 0 for (x, t) ∈ Σ := ∂Ω × (0, T ), (2) u(x, 0) = u0(x) for (x, t) ∈ Ω × {0} (3) where Ω ⊂ R, n = 1, 2, 3, is a bounded Lipschitz domain, and T > 0. Let TN be a decomposition of the space–time cylinder Q = Ω × (0, T ) ⊂ R into simplices τk of mesh size h. For simplicity we assume that the space time cylinder Q has a polygonal (n = 1), a polyhedral (n = 2), or a polychoral (n = 3) boundary ∂Q. With IN we denote the set of all interfaces (interior elements) e between two neighbouring elements τk and τl. For an admissible decomposition the interior elements are edges (n = 1), triangles (n = 2), or tetrahedrons (n = 3). With respect to an interior element e ∈ IN we define for a function v the jump Martin Neumüller · Olaf Steinbach Institute of Computational Mathematics, TU Graz, Steyrergasse 30, 8010 Graz, Austria, e-mail: [email protected],[email protected] 1 2 Martin Neumüller and Olaf Steinbach [v]e(x, t) := v|τk(x, t) − v|τl(x, t) for all (x, t) ∈ e, the average 〈v〉e(x, t) := 1 2 [ v|τk(x, t) + v|τl(x, t) ] for all (x, t) ∈ e, and the upwind in time direction by {v} e (x, t) := { v|τk(x, t) for nt ≥ 0, v|τl(x, t) for nt < 0 for all (x, t) ∈ e, where n = (nx, nt) is the normal vector of the interior element e. For a decomposition TN of the space–time cylinder Q we introduce the discrete function space of piecewise polynomials of order p S h,0(TN ) := { v : v|τk ∈ Pp(τk) for all τk ∈ TN and v|Σ = 0 } . The proposed space–time approach is based on the use of an interior penalty Galerkin approximation of the Laplace operator and an upwind scheme for the approximation of the time derivative, see, e.g., [3, 5]. Hence we have to find uh ∈ S p h,0(TN ) such that aDG(uh, vh) := − N ∑