Algebraic multigrid preconditioner for statically condensed systems arising from lowest-order hybrid discretizations - Di Pietro, Daniele; Hülsemann, Frank; Matalon, Pierre; Mycek, Paul; Rüde, Ulrich (2021)
We address the numerical solution of linear systems arising from the hybrid dis-cretizations of second-order elliptic partial differential equations (PDEs). Such discretizations hinge on a hybrid set of degrees of freedom (DoFs), respectively defined in cells and faces, which naturally gives rise to a global hybrid system of linear equations. Assuming that the cell unknowns are only locally coupled, they can be efficiently eliminated from the system, leaving only face unknowns in the resulting Schur complement, also called statically condensed matrix. We propose in this work an algebraic multigrid (AMG) preconditioner specifically targeting condensed systems corresponding to lowest order discretizations (piecewise constant). Like traditional AMG methods, we retrieve geometric information on the coupling of the DoFs from algebraic data. However, as the condensed matrix only gives information on the faces, we use the uncondensed version to reconstruct the con-nectivity graph between elements and faces. An aggregation-based coarsening strategy mimicking a geometric coarsening or semi-coarsening can then be set up to build coarse levels. Numerical experiments are performed on diffusion problems discretized by the Hybrid High-Order (HHO) method at the lowest order. Our approach uses a K-cycle to precondition an outer flexible Krylov method. The results demonstrate similar performances, in most cases, compared to a standard AMG method, and a notable improvement on anisotropic problems with Cartesian meshes.
Keywords: Satoc condensation; Hybrid methods; Algebraic multigrid

An H-Multigrid Method for Hybrid High-Order Discretizations - Di Pietro, Daniele A.; Hülsemann, Frank; Matalon, Pierre; Mycek, Paul; Rüde, Ulrich; Ruiz, Daniel (SIAM Journal on Scientific Computing, 2021)
We consider a second-order elliptic PDE discretized by the hybrid high-order method, for which globally coupled unknowns are located at faces. To efficiently solve the resulting linear system, we propose a geometric multigrid algorithm that keeps the degrees of freedom on the faces at every grid level. The core of the algorithm lies in the design of the prolongation operator that passes information from coarse to fine faces through the reconstruction of an intermediary polynomial of higher degree on the cells. High orders are natively handled by the use of the same polynomial degree at every grid level. The proposed algorithm requires a hierarchy of nested meshes, such that the faces (and not only the elements) are successively coarsened. Numerical tests on homogeneous and heterogeneous diffusion problems show fast convergence, scalability in the mesh size and polynomial order, and robustness with respect to heterogeneity of the diffusion coefficient.
Keywords: partial differential equations; hybrid high-order; multigrid; static condensation

High-order multigrid strategies for hybrid high-order discretizations of elliptic equations - Di Pietro, Daniele A.; Matalon, Pierre; Mycek, Paul; Rüde, Ulrich (Numerical Linear Algebra With Applications, 2022)
This study compares various multigrid strategies for the fast solution of elliptic equations discretized by the hybrid high-order method. Combinations of (Formula presented.) -, (Formula presented.) -, and (Formula presented.) -coarsening strategies are considered, combined with diverse intergrid transfer operators. Comparisons are made experimentally on 2D and 3D test cases, with structured and unstructured meshes, and with nested and non-nested hierarchies. Advantages and drawbacks of each strategy are discussed for each case to establish simplified guidelines for the optimization of the time to solution.
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Keywords: coarsening strategy; elliptic partial differential equation; hybrid high-order; multigrid

Towards robust, fast solutions of elliptic equations on complex domains through hybrid high-order discretizations and non-nested multigrid methods - Di Pietro, Daniele A.; Hulsemann, Frank; Matalon, Pierre; Mycek, Paul; Rüde, Ulrich; Ruiz, Daniel (International Journal for Numerical Methods in Engineering, 2021)
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 (Formula presented.) -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 (Formula presented.) -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.
CRIS-Team Scopus Importer:2021-08-27
Keywords: hybrid high-order; L2-orthogonal projection; multigrid; non-nested meshes; partial differential equations