An alternating explicit–implicit domain decomposition method for the parallel solution of parabolic equations

Parallel Domain Decomposition Methods for Parabolic Equations

Domain Decomposition Operator Splittings for the Solution of Parabolic Equations

We study domain decomposition counterparts of the classical alternating direction implicit (ADI) and fractional step (FS) methods for solving the large linear systems arising from the implicit time stepping of parabolic equations. In the classical ADI and FS methods for parabolic equations, the elliptic operator is split along coordinate axes; they yield tridiagonal linear systems whenever a un...

An optimization-based domain decomposition method for parabolic equations

An optimization-based, non-overlapping domain decomposition method for the solution of the heat equation is presented. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains. The existence of an optimal solution is proved as is convergence of optimal solutions. A...

Efficient Parallel Solution of Nonlinear Parabolic Partial Differential Equations by a Probabilistic Domain Decomposition

Initialand initial-boundary value problems for nonlinear one-dimensional parabolic partial differential equations are solved numerically by a probabilistic domain decomposition method. This is based on a probabilistic representation of solutions by means of branching stochastic processes. Only few values of the solution inside the space-time domain are generated by a Monte Carlo method, and an ...

A Space Decomposition Method for Parabolic Equations

A convergence proof is given for an abstract parabolic equation using general space decomposition techniques. The space decomposition technique may be a domain decomposition method, a multilevel method, or a multigrid method. It is shown that if the Euler or Crank-Nicolson scheme is used for the parabolic equation, then by suitably choosing the space decomposition, only O(jlog j) steps of itera...