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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.
Source: §1 · p. 2, §2.4 · p. 7Method
The paper projects onto a fully discontinuous broken finite element space and averages the discrete result to restore the required continuity across interfaces.
Source: §1 · p. 2Results
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.
Source: §1 · p. 1, §2.4 · p. 7, §3.1 · p. 8Takeaways & Limitations
For vector-valued elements, the construction shows that commuting bounded cochain projections converge optimally for low-regularity solutions.
Source: §1 · pp. 1–2The 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.
Source: §1 · p. 2, §2.5 · p. 7
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.Source: §1 · p. 1
- 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.Source: §1 · pp. 1–2
- 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.Source: §1 · p. 2
- 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.Source: §1 · p. 2
- 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.Source: §1 · p. 2
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.Source: §2.1 · pp. 2–3
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.Source: §2.2 · pp. 3–4
- 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).Source: §2.2 · pp. 4–5
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.Source: §2.3 · p. 5
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.Source: §2.4 · p. 7, §2.5 · p. 7
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.Source: §3 · p. 8, §3.1 · p. 8
- 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.Source: §3.1 · p. 8
- 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.Source: §3.1 · p. 8
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.Source: §4 · p. 10, §4.1 · p. 10
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.Source: §4.1 · p. 10
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.Source: §4.1 · p. 10
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.Source: §4.2 · p. 11
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.Source: §4.2 · pp. 11–12
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.Source: §5 · p. 12
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.Source: §5 · p. 12
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.Source: §5 · p. 12
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 = ∞.Source: §5 · p. 13
- 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.Source: §5 · pp. 13–14
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.Source: §6 · p. 15, §6.1 · p. 15
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.Source: §6.1 · p. 15, §6.2 · p. 16, §6.3 · p. 17
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.Source: §6.2 · p. 16
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.Source: §6.2 · p. 16, §6.3 · p. 17
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.Source: §6.2 · p. 16
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.Source: §6.3 · pp. 17–18
- 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.Source: §6.3 · pp. 17–18
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.Source: §7 · p. 19
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,∞).Source: §7 · p. 19
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|.Source: §7 · p. 19
- 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.Source: §7 · pp. 19–20