Multi scale preconditioner to solve elliptic problems with high-order methods on non conforming meshes with p4est

This thesis presents a multi scale preconditioner to efficiently solve elliptic problems on unstructured non conforming quadrilateral meshes. Elliptic problems are somewhat limited but they allow for a theoretical understanding and they can be used to build more complex models. The third-party library p4est is used here to handle the dynamic management of the mesh with a forest of quadtrees. This library, that allows for fast adaptive mesh refinement, is designed to work in parallel and scales to hundred of thousands of processor cores. Typically, the number of iterations for typical iterative methods depends on the size of the smallest element and the degree of the interpolation used. When different behaviors occur in the domain of interest, it would be interesting to have elements of different sizes while still being able to compute the solution efficiently. Moreover, reducing the number of unknowns is often a good idea in terms of computation time and memory requirements. Because the refinement of the grid is performed dynamically, the presence of hanging nodes can arise and therefore the proposed preconditioner should handle non conforming meshes, which can sometimes be difficult for traditional methods. In this thesis, a two-scale preconditioner is suggested and implemented from scratch. Its efficiency is then evaluated. The fine part of the preconditioner consists of an overlapping additive Schwarz preconditioner where we solve local problems analytically using some assumption. The second part of the preconditioner is a coarse scale correction where a low order problem is solved by the geometric multigrid method. The performances of the resulting preconditioner are then quantified with several tests. We can observe an h-independent convergence. The number of hanging nodes in the mesh has also no influence on the number of iterations of PCG. The importance of using high-order methods when we have high accuracy requirements is also showed.