Nodal Interpolation Between First-Order Finite Element Spaces in 1D is Uniformly H 1-Stable

Dickopf, Thomas

doi:10.1007/978-3-642-33134-3_45

Cited by 4 publications

(3 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We emphasize that the symbols ⊂ and ⊃ always include the case of equality. The proof and more details are elaborated in [31]. There, we also give counterexamples for the nodal interpolation in higher order finite element spaces.…”

Section: On the H 1 -Stability Of The Nodal Interpolationmentioning

confidence: 96%

Evaluating Local Approximations of the L2-Orthogonal Projection Between Non-Nested Finite Element Spaces

Krause¹

2014

NMTMA

View full text Add to dashboard Cite

We present quantitative studies of transfer operators between finite element spaces associated with unrelated meshes. Several local approximations of the global L 2 -orthogonal projection are reviewed and evaluated computationally. The numerical studies in 3D provide the first estimates of the quantitative differences between a range of transfer operators between non-nested finite element spaces. We consider the standard finite element interpolation, Clément's quasi-interpolation with different local polynomial degrees, the global L 2 -orthogonal projection, a local L 2 -quasi-projection via a discrete inner product, and a pseudo-L 2 -projection defined by a Petrov-Galerkin variational equation with a discontinuous test space. Understanding their qualitative and quantitative behaviors in this computational way is interesting per se; it could also be relevant in the context of discretization and solution techniques which make use of different non-nested meshes. It turns out that the pseudo-L 2 -projection approximates the actual L 2 -orthogonal projection best. The obtained results seem to be largely independent of the underlying computational domain; this is demonstrated by four examples (ball, cylinder, half torus and Stanford Bunny).

show abstract

Section: On the H 1 -Stability Of The Nodal Interpolationmentioning

confidence: 96%

Evaluating Local Approximations of the L2-Orthogonal Projection Between Non-Nested Finite Element Spaces

Krause¹

2014

NMTMA

View full text Add to dashboard Cite

show abstract

“…Applications comprise topics as diverse as a priori and a posteriori error estimates on the one side and multilevel preconditioners on the other side; see [1,2] for a collection of references. The present note is about piecewise linear interpolation (I), which plays a prominent role due to its simplicity.…”

Section: Introductionmentioning

confidence: 99%

“…Note that, beyond the present considerations, the setting S m (I, x) → S n (J, y) for intervals I = J is also studied in [1]; the operator enforcing zero function values at the boundary of the target domain can be treated, too.…”

Section: Introductionmentioning

confidence: 99%

A note on the W^1,p‐stability of piecewise linear interpolation

Dickopf

2012

Proc Appl Math and Mech

View full text Add to dashboard Cite

It is known that piecewise linear interpolation of functions of one variable is uniformly bounded with an H 1 -stability constant of one. In [1], we considered the nodal interpolation operator acting between spaces of piecewise linear functions and presented an elementary proof by minimizing a functional representing the H 1 -semi-norm. In this note, a standard approximation argument is applied generalizing the result to piecewise linear interpolation of all functions in W 1,p on a real interval, 1 ≤ p ≤ ∞. We also comment on alternative proofs and finally give a counterexample for piecewise linear interpolation in 2D.

show abstract

The influence of boundary approximation on transfer operators between non‐nested meshes

Dickopf

Krause

2013

Proc Appl Math and Mech

View full text Add to dashboard Cite

Finite element methods with non-matching meshes can offer increased flexibility in many applications. Although the specific reasons for the use of non-matching meshes are apparently diverse, the common difficulty in all these numerical methods is the transfer of finite element approximation associated with one mesh to finite element approximation associated with another mesh. This paper complements previous quantitative studies of transfer operators between finite element spaces associated with unrelated meshes (T. Dickopf, R. Krause, Evaluating local approximations of the L 2 -orthogonal projection between non-nested finite element spaces, Tech. Rep. 2012-01, Institute of Computational Science, Università della Svizzera italiana, 2012). We study the important use case in which functions are mapped between a regular background mesh and an unstructured mesh of a complex geometry. Here, the former does not approximate the latter.

show abstract

Nodal Interpolation Between First-Order Finite Element Spaces in 1D is Uniformly H 1-Stable

Cited by 4 publications

References 7 publications

Evaluating Local Approximations of the <em>L</em><sup>2</sup>-Orthogonal Projection Between Non-Nested Finite Element Spaces

Evaluating Local Approximations of the <em>L</em><sup>2</sup>-Orthogonal Projection Between Non-Nested Finite Element Spaces

A note on the W^1,p‐stability of piecewise linear interpolation

The influence of boundary approximation on transfer operators between non‐nested meshes

Contact Info

Product

Resources

About

Nodal Interpolation Between First-Order Finite Element Spaces in 1D is Uniformly H 1-Stable

Cited by 4 publications

References 7 publications

Evaluating Local Approximations of the <em>L</em><sup>2</sup>-Orthogonal Projection Between Non-Nested Finite Element Spaces

Evaluating Local Approximations of the <em>L</em><sup>2</sup>-Orthogonal Projection Between Non-Nested Finite Element Spaces

A note on the W1,p‐stability of piecewise linear interpolation

The influence of boundary approximation on transfer operators between non‐nested meshes

Contact Info

Product

Resources

About

A note on the W^1,p‐stability of piecewise linear interpolation