We consider the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic problems can be decoupled via the introduction of a new unknown, corresponding to the boundary value of the solution of the first Laplacian problem. This technique yields a global linear problem that can be solved iteratively via a Krylov-type method. More precisely, at each iteration of the scheme, two second-order elliptic problems have to be solved, and a normal derivative on the boundary has to be computed. In this work, we specialize this scheme for the HHO discretization. To this aim, an explicit technique to compute the discrete normal derivative of an HHO solution of a Laplacian problem is proposed. Moreover, we show that the resulting discrete scheme is well-posed. Finally, a new preconditioner is designed to speed up the convergence of the Krylov method. Numerical experiments assessing the performance of the proposed iterative algorithm on both two- and three-dimensional test cases are presented.
The experiments can be reproduced with the open-source code fhhos4, release 1.2.
Fig. 2
-pb diff -geo square -source sine -s ch -mesh {cart|tri} -normalder -k {0|1|2|3} -n {16|32|64|128|256|512}
Fig. 3, 4, 5
# Cartesian mesh
-pb bihar -geo square -source {exp|poly} -s ch -bihar-prec s -nbh-depth 8 -mesh cart -cs r -k {0|1|2|3} -n {16|32|64|128|256} -tol 1e-10
# Polygonal mesh
-pb bihar -geo square -source exp -s ch -bihar-prec s -nbh-depth 8 -bihar-prec-solver bicgstab -mesh poly -polymesh-init cart -polymesh-n-pass 1 -polymesh-fcs c -k {0|1|2|3} -n {16|32|64|128|256} -tol 1e-10
Fig. 7
-pb bihar -geo square -source poly -s ch -mesh cart -cs r -k 1 -n 256 -tol 1e-14 -export iter -bihar-prec no #no preconditioner
-pb bihar -geo square -source poly -s ch -mesh cart -cs r -k 1 -n 256 -tol 1e-14 -export iter -nbh-depth {2|4|6|8|10} #with preconditioner
Tables 1, 2, 3
-pb bihar -geo square -source exp -mesh cart -cs r -s ch -nbh-depth 8 -tol 1e-8 -bihar-prec {s|no} -k {0,1,2,3} -n {32,64,128,256,512}
-pb bihar -geo cube -source exp -mesh tetra -not-compute-errors -s fcguamg -hp-cs p_h -nbh-depth 2 -bihar-prec-solver bicgstab -tol 1e-8 -k 0 -bihar-prec {s|no} -n {8,16,32,64}