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Finite element quasi-interpolation and best approximation
Alexandre Ern, Jean-Luc Guermond
TL;DR
Finite element best-approximation analysis needs operators that remain effective for low-regularity scalar and vector-valued functions across conforming spaces. The paper constructs a two-step projection-and-averaging quasi-interpolant with L1 stability, point-wise invariance, and optimal approximation over fractional Sobolev spaces, including H1, H(curl), and H(div) settings. The framework also clarifies that vector-valued quasi-interpolation complements commuting projections and supports their optimal convergence for low-regularity solutions.
Problem
Canonical interpolation operators for the finite element examples are not stable in Lp spaces, while quasi-interpolation results for H(curl)- and H(div)-conforming elements were missing.
Method
The paper projects onto a fully discontinuous broken finite element space and averages the discrete result to restore the required continuity across interfaces.
Results
The operator is L1-stable, point-wise invariant, and optimally approximates W^{r,p} functions for p ∈ [1, ∞] and r arbitrarily close to zero, with or without homogeneous boundary conditions.
Takeaways & Limitations
For vector-valued elements, the construction shows that commuting bounded cochain projections converge optimally for low-regularity solutions.
Takeaways & Limitations
The framework assumes degrees of freedom satisfy estimates relating them to mesh-interface jumps and traces, and its stated literature gap concerns H(curl)- and H(div)-conforming quasi-interpolation.
Abstract
from arXiv · showhide
This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces.This operator gives optimal estimates of the best approximation error in any $L^p$-norm assuming regularity in the fractional Sobolev spaces $W^{r,p}$, where $p\in [1,\infty]$ and the smoothness index $r$ can be arbitrarily close to zero. The operator is stable in $L^1$, leaves the corresponding finite element space point-wise invariant whether homogeneous boundary conditions are imposed or not. The theory is illustrated on $H^1$-, $\mathbf{H}(\text{curl})$- and $\mathbf{H}(\text{div})$-conforming spaces.
1. Introduction.
The paper constructs a quasi-interpolation framework for scalar- and vector-valued finite element spaces that is L1-stable, point-wise invariant, and optimally approximates low-regularity functions. Its two-step projection-and-averaging route extends the discussion to H1-, H(curl)-, and H(div)-conforming elements, where prior quasi-interpolation results were missing for the latter two settings.
- 1. Introduction.: The operator is stable in L1, point-wise invariant on the finite element space, and supports optimal approximation for W^{r,p} functions with r arbitrarily close to zero.The construction also accommodates homogeneous boundary conditions legitimate in W.
- 1. Introduction.: For vector-valued elements, the construction complements rather than replaces commuting bounded cochain projections.Its approximation results imply optimal convergence of such projections for low-regularity solutions.
- 1. Introduction.: Prior quasi-interpolation results covered H1-conforming elements, whereas corresponding results for H(curl)- and H(div)-conforming elements were missing from the literature.The cited H1 developments include L1-stable and fractional-regularity extensions of Scott–Zhang interpolation.
- 1. Introduction.: The key construction first projects onto a fully discontinuous broken finite element space and then averages the discrete result to stitch continuity across interfaces.Mesh-cell patches are used only when handling discrete objects, and the analysis applies across different types of degrees of freedom.
- 1. Introduction.: The framework targets affine, shape-regular meshes and finite element spaces with prescribed continuity across mesh interfaces.The paper develops the abstract spaces, local interpolation, averaging, quasi-interpolation, boundary conditions, and fractional Sobolev estimates in successive sections.
2. Finite elements.
The paper formalizes affine, shape-regular finite element families on matching meshes and identifies conforming spaces through interface continuity conditions. These spaces support scalar and vector-valued elements, including Lagrange, Nédélec, and Raviart–Thomas types.
- Meshes: Affine, shape-regular mesh sequences partition a bounded Lipschitz polyhedron into matching convex cells related to a reference element.Each cell is obtained from the reference element by an affine bijection.
- Finite element generation: A finite element is generated from a reference triple by transporting its polynomial space, degrees of freedom, and shape functions to each mesh cell.The resulting local degrees of freedom remain unisolvent, and the local space is point-wise invariant under canonical interpolation.
- Finite element generation: Affine pullback and pushforward maps provide uniform Sobolev-norm bounds whose constants depend on mesh shape regularity and derivative order.The corresponding operator composition estimate scales like h^(s-m).
- Abstract finite element spaces: The global conforming space P(Th) is the subspace of the broken finite element space characterized by zero γ-jumps across mesh interfaces.The framework allows selected trace components to define conformity in a functional space W.
- Finite element examples: The theory covers scalar Lagrange elements and vector-valued Nédélec- and Raviart–Thomas-type elements with interface continuity encoded through degrees of freedom.The paper targets quasi-interpolation operators stable in L1 with optimal approximation properties, with or without homogeneous boundary conditions.
3. L1-stable local interpolation.
The local interpolation construction extends finite element degrees of freedom to integrable functions, producing an L1-stable operator that commutes with pullback and preserves the local finite element space.
- Extension of degrees of freedom: The extended local degrees of freedom allow interpolation of functions that are only integrable.They are constructed through bounded reference-element functionals and the element pullback.
- Extension of degrees of freedom: The local operator I♯_K is uniformly stable in Lp for every p ∈ [1, ∞].The proposition states a uniform operator bound on each mesh cell.
- Extension of degrees of freedom: I♯_K commutes with the element pullback and leaves the local finite element space point-wise invariant.These properties preserve the reference-to-physical-element structure while retaining exactness on finite element functions.
4. Averaging operator.
The averaging operator maps the broken finite element space into the conforming space by averaging connectivity classes of local shape-function data. It is locally stable and admits approximation estimates.
- Operator construction: J_h^av is a bounded linear averaging operator from the broken finite element space P̄(Th) to the conforming space P(Th).Its construction uses global shape functions and their connectivity classes across cells.
- Connectivity: Connectivity sets collect all cell-local degrees of freedom associated with one global shape function, including those joined across interfaces.Singleton connectivity classes correspond to shape functions supported on one element.
- Interface control: The averaging analysis relies on a uniform relation between degree-of-freedom differences and γ-jumps across interfaces.This assumption is stated for the H1-, H(curl)-, and H(div)-type finite elements considered in the paper.
- Stability: The averaging operator is locally Lp-stable for all p ∈ [1,∞].The result is stated uniformly over broken finite element functions and mesh cells.
- Approximation: A local approximation estimate holds for the averaging error for all m ∈ {0:k+1} and p,r ∈ [1,∞].The proof controls the difference between broken and averaged functions through interface jumps, paths of cells, and inverse inequalities.
5. Quasi-interpolation operator.
The paper constructs a globally assembled quasi-interpolation operator that is stable, point-wise invariant on the finite element space, and supports local and global approximation estimates in fractional Sobolev spaces.
- Construction: The global operator is constructed by averaging locally projected finite element functions across mesh interfaces.The approximation process first uses a local operator and then an averaging operator to assemble a conforming result.
- Operator properties: The operator is point-wise invariant on P(Th), so applying it to any finite element function reproduces that function.This invariance is used in the approximation argument and is also stated for the boundary-condition variant.
- Stability: The operator satisfies W^m,p-stability estimates for p ∈ [1, ∞] and derivative orders m from 0 through k + 1.The stated stability applies to functions on element patches DK and has a uniform constant.
- Approximation: Local approximation estimates hold for r ∈ [0, k + 1], including fractional smoothness with r arbitrarily close to zero.For noninteger r, the stated range of p is [1, ∞); for integer r, it extends to p = ∞.
- Approximation: The resulting global estimate bounds the W^m,p-seminorm error by the best polynomial approximation error over the mesh.The global best-approximation result is stated for the same fractional and integer smoothness ranges as the local theorem.
- Boundary conditions: A variant of the construction prescribes homogeneous boundary values when those conditions are legitimate in W.The paper introduces this variant as a separate goal after constructing the unconstrained quasi-interpolation operator.
6. Quasi-interpolation with boundary prescription.
This section constructs a boundary-aware quasi-interpolation operator and establishes its stability and approximation properties, including regimes below and above the trace threshold rp=1.
- Operator construction: The construction modifies an averaging operator to prescribe homogeneous boundary conditions while retaining a global quasi-interpolation operator.The framework distinguishes internal and boundary degrees of freedom and boundary cells to define the operator.
- Stability: The operator is Lp-stable uniformly for all p∈[1,∞] on the relevant finite element spaces.The stability lemma applies to all v∈P^b(Th) and all mesh cells K.
- Approximation: The averaging operator provides local approximation estimates for derivatives up to the available polynomial degree and fractional Sobolev regularity.The estimate covers m∈{0,…,k+1}, p,r∈[1,∞], with the stated restrictions on p when r is noninteger.
- Invariance: The boundary-prescribed operator preserves point-wise invariance of the homogeneous finite element space and acts as a projection there.This follows from the corresponding invariance of the modified averaging operator.
- Global estimates: For rp>1, global boundary-cell estimates extend the approximation result to functions satisfying the homogeneous trace condition.The constant may depend on |rp−1| in this regime.
- Global estimates: For rp<1, the global estimate controls boundary behavior but cannot improve the mismatch caused by forcing the interpolant to vanish at the boundary.The paper notes that v may blow up like ρ^−s_w near the boundary while I_av_h0(v) is zero there.
7. Technical results in fractional Sobolev spaces.
This section supplies fractional Poincaré and trace inequalities used to derive local estimates on affine, shape-regular meshes.
- Technical tools: The section develops two technical tools in fractional Sobolev spaces: a Poincaré inequality and a trace inequality.These results support the approximation analysis for fractional regularity.
- Poincaré inequality: The fractional Poincaré inequality bounds deviations from the average on an open set using its diameter and fractional Sobolev seminorm.It applies for s∈(0,1) and p∈[1,∞).
- Trace inequality: The trace inequality applies when sp>1 for fractional regularity, or when s=1, with constants uniform over the mesh sequence.For fractional s, the constant depends on |sp−1|.
- Trace inequality: Affine changes of variables and mesh shape-regularity transfer the trace estimate from a reference cell to each physical mesh cell.The proof explicitly uses the affine map and uniform shape-regularity.