Grant: Discovery Grants Program - Individual

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. We consider PDEs 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 track 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. Moreover, we wish to study algorithms specifically designed to take advantage of readily available compute clusters or cloud computing with hundreds or thousands of 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 algorithms. We will provide implementations and analyze algorithms for the generation of the adaptive grids and the solution of the physical PDE by solving the coupled mesh-physical  system in either an alternating or monolithic framework. Software for the adaptive, parallel solution of PDEs on surfaces (suitable for high performance computing (HPC) environments) will be developed and made available on Github (or similar), and components made available to support existing high quality PDE solvers such as FireDrake and Fenics.  The research program will also study two applications: the spread of wildfires and the simulation of corrosion processes. These problems are of huge societal and industrial interest and their numerical solution would benefit greatly from the use of adaptive meshing and parallel computing.  This work will be completed with experts in the natural resources and engineering fields.  Highly qualified personnel trained during this research program will be equipped to continue in research positions or academic careers in numerical analysis and scientific computing, trained in the principles of equity, diversity, and inclusion (EDI), and will gain communication and computing skills which are transferable to industry and positions in the burgeoning tech sectors.