Towards robust, fast solutions of elliptic equations on complex domains through hybrid high‐order discretizations and non‐nested multigrid methods

Abstract

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 2orthogonal 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 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.

High‐order multigrid strategies for hybrid high‐order discretizations of elliptic equations

“…The prolongation operator employed in our previous works [12,13] in the context of h-multigrid is here extended to the cases where k > k −1 and/or M −1 = M , in order to handle p-and hp-coarsening as well.…”

“…We refer the reader to [12,13] for more details about these operators. The prolongation operator by decondensation P :…”

“…Once the potential v T ∈ P k (T) is retrieved from (10) via its algebraic representation, we improve its accuracy through the higher-order reconstruction operator p k+1 T defined by (3). The local contribution to Θ k 𝓁 is therefore given by…”

“…Among them, one can cite the discontinuous Petrov–Galerkin method 3,4 and the hybridizable discontinuous Galerkin method 5‐8 . Regarding HHO, an $$$ h $$$‐multigrid method was designed by the present authors for the diffusion equation in Reference 9, extended to non‐nested mesh hierarchies in Reference 10, and a $$$ p $$$‐multigrid method for the Stokes equation was proposed in Reference 11.…”

“…This work follows up on References 9 and 10, where an efficient $$$ h $$$‐multigrid method was designed. The convergence of the algorithm, which preserves the polynomial degree at every multigrid level, is shown to be independent of the mesh size (and weakly dependent on the polynomial degree).…”