Domain splitting algorithm for mixed finite element approximations to parabolic problems

In this paper we formulate and study a domain decomposition algorithm for solving the mixed finite element approximations to parabolic initial-boundary value problems. In contrast to the ordinary overlapping domain decomposition method this technique leads to noniterative algorithms, i.e. the subdomain problems are solved independently and the solution in the whole domain is obtained from the local solutions by restriction and simple averaging. The algorithm implies that the time discretization leads to an elliptic problem with a large positive coefficient in front of the zero order term. The solutions of such problems exhibit a boundary layer with thickness proportional to the square root of the time discretization parameter. Thus, any error in the boundary conditions decays exponentially and a reasonable overlap produces a sufficiently accurate method. We prove that the proposed algorithm is stable in L2-norm and has the same accuracy as the implicit method.

Domain splitting algorithm for mixed finite element approximations to parabolic problems Academic Article