We prove the uniform convergence of the geometric multigrid V-cycle for hybrid high-order and other discontinuous skeleton methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Our generalized framework allows for injection operators that are local to each coarse grid cell. Numerical experiments confirm our theoretical results.
The experiments can be reproduced with the open-source code fhhos4, release 1.2.1.
This injection operator is enabled by the parameters -prolong 2 -disable-hor.
-pb diff -geo square -mesh stri -mesher inhouse -s mg -cs s -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 2 -disable-hor -k {1|2|3} -n {32|64|128|256|512}
This injection operator is enabled by the parameters -prolong 1 -disable-hor.
-pb diff -geo square -mesh stri -mesher inhouse -s mg -cs s -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -disable-hor -k {1|2|3} -n {32|64|128|256|512}
This injection operator is enabled by the parameter -prolong 1.
-pb diff -geo square -mesh stri -mesher inhouse -s mg -cs s -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -k {1|2|3} -n {32|64|128|256|512}
-pb diff -geo L_shape -mesh tri -mesher gmsh -s mg -cs r -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 2 -disable-hor -k {1|2|3} -n {16|32|64|128|256}
-pb diff -geo L_shape -mesh tri -mesher gmsh -s mg -cs r -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -disable-hor -k {1|2|3} -n {16|32|64|128|256}
-pb diff -geo L_shape -mesh tri -mesher gmsh -s mg -cs r -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -k {1|2|3} -n {16|32|64|128|256}
-pb diff -geo cube -mesh stetra -mesher inhouse -s mg -cs s -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -k {1|2|3} -n {8|16|32}