Multidomain local Fourier method for PDEs in complex geometries

A low communication parallel algorithm is developed for the solution of time-dependent nonlinear PDEs. The parallelization is achieved by domain decomposition. The discretization in time is performed via a third-order semi-implicit stiffly stable scheme. The elemental solutions in the subdomains are constructed using a high-order method with the local Fourier basis (LFB). The continuity of the global solution is accomplished by a point-wise matching of the local subsolutions on the interfaces. The matching relations are derived in terms of the jumps on the interfaces. The LFB method enables splitting a two-dimensional problem with global coupling of the interface unknowns into a set of uncoupled one-dimensional differential equations. Localization properties of an elliptic operator, resulting from the discretization in time of a time-dependent problem, are utilized in order to simplify the matching relations. In effect, only local (neighbor- to -neighbor) communication between the processors becomes necessary. The present method allows the treatment of problems in various complex geometries by the mapping of curvilinear domains into simpler (rectangular or circular) regions with subsequent matching of local solutions. The operator with nonconstant coefficients, obtained in the transformed domain, is preconditioned by an appropriate constant coefficient operator, easily inverted by the LFB. The problem is then solved with spectral accuracy by (a rapidly convergent) conjugate gradient iteration.