Conditioning of a Hybrid High-Order Scheme on Meshes with Small Faces

Abstract

We conduct a condition number analysis of a Hybrid High-Order (HHO) scheme for the Poisson problem. We find the condition number of the statically condensed system to be independent of the number of faces in each element, or the relative size between an element and its faces. The dependence of the condition number on the polynomial degree is tracked. Next, we consider HHO schemes on cut background meshes, which are commonly used in unfitted discretisations. It is well known that the linear systems obtained on these meshes can be arbitrarily ill-conditioned due to the presence of sliver-cut and small-cut elements. We show that the condition number arising from HHO schemes on such meshes is not as negatively effected as those arising from conforming methods. We describe how the condition number can be improved by aggregating ill-conditioned elements with their neighbours.

Stability and Conditioning of Immersed Finite Element Methods: Analysis and Remedies

“…This method bypasses the aforementioned three computational challenges of immersed finite element methods, but instead introduces other challenges, such as a non-trivial treatment of boundary conditions (including the projection of the boundary data), and a non-obvious geometrical treatment. Another approach is to use hybridizable techniques on unfitted meshes [94,95]. Hybridizable methods can naturally be posed on polytopal meshes, giving additional geometrical flexibility compared to standard finite element methods.…”

“…As a result, these methods can readily be used on the meshes obtained after the intersection of the boundary representation and background mesh and possibly after aggregation of elements. The impact of small cut elements and small cut faces (these schemes add unknowns on the mesh skeleton) on stability and condition numbers has only been studied very recently in the context of hybrid highorder methods, see [95]. A detailed discussion of these alternative techniques is beyond the scope of this work.…”

An Arbitrarily High Order Unfitted Finite Element Method for Elliptic Interface Problems with Automatic Mesh Generation

“…The second objective of the paper is to study the condition number of the stiffness matrix of high-order unfitted finite element methods which are known to be of the order O(h −2 ) in the literature [16,33,32,9,6] on quasi-uniform meshes with the mesh size h. For high order methods, it is known [44] that the condition number of the stiffness matrix may grow exponentially with the finite element approximation order p in terms of the measure of cut cells. This indicates that the geometry of the cut cells is essential in controlling the condition number of the stiffness matrix.…”

Stability and conditioning of immersed finite element methods: analysis and remedies

“…This method bypasses the aforementioned three computational challenges of immersed finite element methods, but instead introduces other challenges, such as a non-trivial treatment of boundary conditions (including the projection of the boundary data), and a non-obvious geometrical treatment. Another approach is to use hybridizable techniques on unfitted meshes [91,92]. Hybridizable methods can naturally be posed on polytopal meshes, giving additional geometrical flexibility compared to standard finite element methods.…”

“…As a result, these methods can readily be used on the meshes obtained after the intersection of the boundary representation and background mesh and possibly after aggregation of elements. The impact of small cut elements and small cut faces (these schemes add unknowns on the mesh skeleton) on stability and condition numbers has only been studied very recently in the context of hybrid high-order methods, see [92]. A detailed discussion of these alternative techniques is beyond the scope of this work.…”