Toward robust, fast solutions of elliptic equations on complex domains through HHO discretizations and non-nested multigrid methods
Résumé
The use of modern discretization technologies such as Hybrid High-Order (HHO) methods, coupled with appropriate linear solvers, allow for the robust and fast solution of Partial Differential Equations (PDEs). Although efficient linear solvers have recently been made available for simpler cases, complex geometries remain a challenge for large scale problems. To address this problem, we propose in this work a geometric multigrid algorithm for unstructured non-nested meshes. The non-nestedness is handled in the prolongation operator through the use of the $L^2$-orthogonal projection from the coarse elements onto the fine ones. However, as the exact evaluation of this projection can be computationally expensive, we develop a cheaper approximate implementation that globally preserves the approximation properties of the $L^2$-orthogonal projection. Additionally, as the multigrid method requires not only the coarsening of the elements, but also that of the faces, we leverage the geometric flexibility of polytopal elements to define an abstract non-nested coarsening strategy based on element agglomeration and face collapsing. Finally, the multigrid method is tested on homogeneous and heterogeneous diffusion problems in two and three space dimensions. The solver exhibits near-perfect asymptotic optimality for moderate degrees of approximation.
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