Evaluation of the condition number in linear systems arising in finite element approximations

Ern, Alexandre; Guermond, Jean‐Luc

doi:10.1051/m2an:2006006

Cited by 50 publications

(41 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The condition number of G indeed scales in the same way as the condition number of the standard boundary fitted FEM method defined on V k h as we show in the following lemma (see also [4] for a general discussion of condition numbers of finite element matrices): …”

Section: Lemma 42 There Existssupporting

confidence: 53%

Ghost penalty

Burman

2010

Comptes Rendus. Mathématique

343

370

View full text Add to dashboard Cite

In this Note we discuss a simple penalty method that allows to increase the robustness of fictitious domain methods. In particular the condition number of the matrix can be upper bounded independently of how the domain boundary intersects the computational mesh, under rather weak assumptions. r é s u m éDans cette Note nous étudions une méthode de pénalisation simple pour des méthodes de domaine fictif. La méthode permet de contrôler la sensibilité du nombre de conditionnement du système linéaire en fonction du positionement du domaine par rapport au maillage.

show abstract

Section: Lemma 42 There Existssupporting

confidence: 53%

Ghost penalty

Burman

2010

Comptes Rendus. Mathématique

343

370

View full text Add to dashboard Cite

show abstract

“…In this section, we formulate two abstract assumptions on the stabilized bilinear form (3.15) which allow us to establish optimal condition number bounds for the associated discrete system which are independent of the position of the manifold Γ relative to the background mesh T h . Following the approach in [15], we require that for v ∈ V h,0 , the discrete bilinear form A h satisfies • a discrete Poincaré estimate…”

Section: Abstract Condition Number Estimatesmentioning

confidence: 99%

“…Again, by a partition of unity argument, we can assume that Γ is given by a single parametrization. Recalling the definition of v e and using tube coordinates (7.1) defined by Φ in combination with the measure equivalence (7.5), the tube integral for l = 0 computes to Next, we recall from [15]…”

Section: An Interpolation Operator: Construction and Estimatesmentioning

confidence: 99%

Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions

et al. 2018

View full text Add to dashboard Cite

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in R d of arbitrary codimension. The method is based on using continuous piecewise polynomials on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in R 3 .

show abstract

“…To prove an upper bound on the stiffness matrix condition number we follow the approach in [3,5]. Let tϕ i u N i"1 be the standard piecewise tensor product polynomial Lagrange basis functions associated with the nodes in K h and let A and M be the stiffness and mass matrices with elements A ij " A h pϕ j , ϕ i q and M ij " pϕ j , ϕ i q Ω , respectively.…”

Section: Condition Number Estimatementioning

confidence: 99%

Cut finite element methods for elliptic problems on multipatch parametric surfaces

Jonsson

Larson

Larsson

2017

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

We develop a finite element method for the Laplace-Beltrami operator on a surface described by a set of patchwise parametrizations. The patches provide a partition of the surface and each patch is the image by a diffeomorphism of a subdomain of the unit square which is bounded by a number of smooth trim curves. A patchwise tensor product mesh is constructed by using a structured mesh in the reference domain. Since the patches are trimmed we obtain cut elements in the vicinity of the interfaces. We discretize the Laplace-Beltrami operator using a cut finite element method that utilizes Nitsche's method to enforce continuity at the interfaces and a consistent stabilization term to handle the cut elements. Several quantities in the method are conveniently computed in the reference domain where the mappings impose a Riemannian metric. We derive a priori estimates in the energy and L 2 norm and also present several numerical examples confirming our theoretical results.Subject Classification Codes: 65M60, 65M85.

show abstract

Evaluation of the condition number in linear systems arising in finite element approximations

Cited by 50 publications

References 12 publications

Ghost penalty

Ghost penalty

Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions

Cut finite element methods for elliptic problems on multipatch parametric surfaces

Contact Info

Product

Resources

About