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A quasi-interpolation method for finite element spaces
Ohannes Karakashian
TL;DR
Approximation analysis can require special treatment for elements that are not affine equivalent, while bounds for affine elements may need seminorms the usual approach does not supply. The paper combines local Taylor polynomials, averaging, and trace estimates; for Courant’s triangle, it gives O(h) and O(h^2) error bounds under stated conditions.
Problem
Some finite elements are not affine equivalent and require special treatment for approximation bounds, while affine elements may lack bounds in the H1 seminorm needed for a posteriori estimation.
Source: §1 · pp. 1–2Method
The method constructs local Taylor polynomial approximations and averages local degrees of freedom, using trace estimates to bound jumps and obtain a global finite-element approximation.
Source: §1 · p. 2Results
For Courant’s triangle, the paper gives an O(h) L2 error bound in terms of the H1 seminorm and an O(h^2) bound for H2 functions.
Source: §3 · p. 10Takeaways & Limitations
The combined approach is presented as widely applicable to practical finite elements and simple to apply to a particular element.
Source: §1 · p. 2Boundary conditions on Vh are not treated in detail; homogeneous conditions are described as relatively easy on polygonal domains, while more general ones can use Nitsche’s method.
Source: §3 · p. 15
Abstract
from arXiv · showhide
A combination of Taylor polynomial approximation and averaging results in a powerful quasi-interpolation technique that can be used to analyze the approximation properties of finite element spaces. With this particular approach, the notion of affine equivalent families is completely bypassed in the sense that the degrees of freedom are defined locally rather than being exported from a reference element. In particular, domains with curved boundaries can be handled in a natural manner. Finally, the ease of its application to a particular finite element makes it a valuable pedagogical tool.
1 Introduction
Standard approximation analyses can require affine-equivalent families and may not provide bounds in the seminorms needed for affine elements. The paper combines cellwise Taylor approximation with averaging and trace estimates to construct finite-element approximations without reference elements.
1 Introduction
- The usual scaling approach relies on a master element, while affine elements may lack bounds in the H1 seminorm needed for a posteriori estimation.Some elements, including those with normal-derivative degrees of freedom, are not affine equivalent and require special treatment to derive approximation bounds.Source: §1 · pp. 1–2
- The approach combines Taylor approximation, trace estimates, and Oswald averaging; its novelty is their combination, rather than the individual tools.The paper presents the combination as widely applicable to practical finite elements and simple to apply to a particular element.Source: §1 · p. 2
- The method builds discontinuous cellwise Taylor polynomials, then averages local degrees of freedom to obtain a function in the global finite-element space.Trace estimates bound jumps between local Taylor polynomials in terms of the cellwise difference from the target function.Source: §1 · pp. 2–3
- The Taylor remainder is estimated directly, without the averaging step used in the cited averaged-Taylor approaches.The paper notes that concentration can prevent the remainder from being optimally bounded at every point.Source: §1 · p. 2
- Local degrees of freedom replace affine exports from reference elements, and Taylor approximation can be used on cells with curved boundaries.The authors suggest that avoiding master elements may open possibilities for meshes mixing cell geometries; global convexity is not required.Source: §1 · pp. 2–3
- The paper presents the tools and applies them to the Courant triangle, Argyris triangle, and Hsieh-Clough-Tocher macro-element.The HCT element is highlighted as a non-polynomial finite element; the paper frames the examples as illustrating the method’s efficacy.Source: §1 · p. 3
2 Preliminaries
The section establishes local approximation and trace tools on starlike, shape-regular cells, then develops averaging estimates for constructing global finite-element functions from local data.
Taylor approximation
- Local Taylor polynomials yield discontinuous piecewise-polynomial approximants with optimal approximation properties in a variety of norms.The paper describes its proof as more direct than the averaged Taylor polynomial development of Dupont and Scott.Source: §2 · p. 4
- For u in H^{k+1}(D) on a starlike, shape-regular D, a degree-k polynomial χ satisfies |u − χ|_{l,D} ≤ c(k + 1)h^{k+1−l}|u|_{k+1,D} for l = 0, ..., k.The constant depends only on n, D, and the ratio µ(D)/µ(S).Source: §2 · p. 4
Trace estimate
- The trace estimate balances the boundary L2 norm against interior L2 and H1-seminorm terms, works for typical cells, and does not adapt well to thin sets.The theorem assumes a starlike D satisfying assumptions (A1) and (A2).Source: §2 · p. 6
Averaging
- For a collection of real numbers, the error from replacing them by their average can be bounded by consecutive differences.The constant in the bound depends only on the number of values.Source: §2 · p. 7
Finite-element cells
- Peripheral mesh cells may have one curved face, provided they are convex and contain an affine cell used to define the nodes; their degrees of freedom remain locally defined.The degrees of freedom are not exported from a reference element.Source: §2 · p. 8
Basis functions
- The canonical basis functions satisfy the proposition’s bounds with a constant independent of the cell diameter.The result concerns finite elements whose degrees of freedom include directional derivatives at nodes.Source: §2 · p. 8
Averaging
- A discontinuous piecewise-polynomial function vh can be averaged into a global χ whose local errors are bounded purely by jumps of directional derivatives across mesh edges.This is the stated cornerstone of the approach; the local polynomial spaces are stitched by averaging degrees of freedom.Source: §2 · p. 9
3 Examples
The examples apply the Taylor-approximation and averaging approach to several finite elements, including elements on meshes with curved peripheral sides. They also show how lower-order Taylor approximations can reduce regularity requirements.
Mesh assumptions
- The examples use shape-regular, locally quasi-uniform triangular meshes, allowing peripheral triangles to have one curved side.Peripheral cells contain a regular triangle used to define nodes.Source: §3 · p. 10
Courant’s triangle
- For Courant’s triangle, the section gives an O(h) L2 error estimate in terms of the H1 seminorm and an O(h2) error bound for u ∈ H2(Ω).The O(h) estimate is described as equivalent in some sense to the Clément interpolant result and as useful for a posteriori estimation.Source: §3 · p. 10
Taylor approximation
- Using Taylor approximations below the finite-element degree gives lower powers of h while requiring lower regularity of u.The section contrasts this option with the Bramble–Hilbert-type approach, which does not allow that feature.Source: §3 · pp. 11–12
Argyris triangle
- For u ∈ C2(Ω) ∩ H6(Th), Theorem 4 asserts that there exists χ ∈ Vh, the globally C1, piecewise-quintic space.The proof constructs χ from the collection of cell-Taylor polynomials of degree 5 on Th by averaging as in Proposition 2.Source: §3 · pp. 11–12
HCT triangle
- The HCT example states an approximation result for u ∈ C1(Ω) ∩ H4(Th) in a globally C1 space of composite cubic elements.Each cell consists of three cubic pieces stitched together in a C1 fashion.Source: §3 · p. 13
Scope
- The section notes that the HCT element does not satisfy the inclusion assumption used by the techniques of for k = 3.Boundary conditions are not treated in detail; homogeneous conditions on polygonal domains are described as relatively easy to incorporate, while more general conditions may use Nitsche’s method.Source: §3 · pp. 14–15