Analysis and Implementation of Parallel Solvers for PDE Based Mesh Generation and Coupled Systems

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Many models of interest to engineers and scientists are written mathematically as partial differential equations (PDEs). Except for certain idealized situations, the resulting PDEs are not possible to solve analytically. Instead we rely on numerical approximations. Of particular interest is the development of efficient implementations and the analysis of adaptive algorithms for the solution of time-dependent PDEs in two or three spatial dimensions, possibly living on non-flat surfaces, whose solutions exhibit large solution variation, singularity formation or moving fronts. My students and I study methods which obtain an efficient solution by automatically concentrating computational effort in space and time regions in which the solution has this interesting, but difficult to track behaviour. The strategy works by using automatically generated moving spatial meshes to adapt to the desired features. This is achieved by formulating a (possibly nonlinear) PDE for the mesh, which depends on the unknown solution, and hence is coupled to the physical PDE. We wish to study algorithms specifically designed to take advantage of readily available compute clusters with hundreds or thousands of cores, hybrid CPU-GPU systems and even desktop machines with multiple cores. We propose mapping the adaptive solution of time dependent PDEs on surfaces to multicore environments by dividing the large problem into small pieces, computing on individual cores and then recombining to give a solution of the original problem using domain decomposition (DD) algorithms and preconditioners. We will provide implementations and analyze algorithms for the generation of the adaptive grids and the solution of the physical PDE in either an alternating or monolithic (one-shot) framework. To saturate very large numbers of cores, small scale parallelism in time will be added by computing simultaneous predictions and corrections. The resulting software will be applied to geophysical electromagnetic problems in scenarios relevant to natural resource exploration. Extending our solvers to PDEs defined on surfaces will allow for fast, scalable simulations on more realistic geometries of interest to computational scientists and engineers.Ultimately, we will provide a new, theoretically based, modular platform for the parallel adaptive solution of time dependent PDEs on general surfaces suitable for existing and emerging high performance computing hardware. This research program provides an impact chain for PDE based mesh generation methods, providing a software tool for computational scientists requiring the solution of complex problems. Theoretically, the research program will enhance our knowledge of the behaviour of DD algorithms for nonlinear coupled systems. Finally, it will provide HQP with mathematical expertise, computational capability, and key transferable skills highly sought by employers.