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Space-time finite element interpolation of nonsmooth functions satisfying boundary conditions
Thomas Führer, Gregor Gantner, Roberto González, Michael Karkulik
TL;DR
The paper addresses quasi-interpolation for nonsmooth functions in parabolic space-time finite elements, with particular attention to discrete Dirichlet boundary conditions. It constructs a stable projection on tensor-product meshes and applies it to obtain a quasi-optimal method for inhomogeneous boundary data.
Problem
Variational PDE solutions may lack the smoothness needed for point evaluation, motivating quasi-interpolation for space-time finite elements.
Source: §1 · p. 1Method
The paper constructs a linear, uniformly bounded projection on tensor-product space-time meshes that preserves discrete lateral boundary traces.
Source: §3.1 · pp. 5–6Results
The resulting method for inhomogeneous Dirichlet data is quasi-optimal, with an error bound that includes boundary-trace mismatch and constrained best approximation.
Source: §2 · p. 4Takeaways & Limitations
The operator provides a space-time finite-element tool for incorporating inhomogeneous Dirichlet data into schemes designed for homogeneous boundary conditions.
Source: §3.2 · p. 8The local stability constants depend on temporal local quasi-uniformity, spatial shape regularity (or local quasi-uniformity in one dimension), and the polynomial degrees.
Source: §3.1 · p. 6
Abstract
from arXiv · showhide
We construct Scott-Zhang-type space-time quasi-interpolation operators on tensor- product meshes for parabolic settings. Specifically, these are linear and uniformly bounded projections from the classical parabolic energy space onto a space-time tensor-product finite element space that preserve discrete Dirichlet boundary conditions on the lateral space-time boundary. We apply our operator in the context of space-time first-order system least-squares finite elements for parabolic equations to establish a quasi-optimal method for inhomogeneous Dirichlet boundary conditions.
1. Introduction
Space-time variational methods have a long history, but their finite-element theory has only recently received thorough investigation. The section motivates quasi-interpolation for nonsmooth PDE solutions and describes the Scott–Zhang construction.
1. Introduction
- Space-time variational formulations have been studied for at least fifty years, while their finite-element theory has only recently been investigated more thoroughly.Classical time-stepping analysis is described as textbook knowledge.Source: §1 · p. 1
- Quasi-interpolation is needed because variational PDE solutions may lack the smoothness required for point evaluation.Source: §1 · p. 1
- For fixed-degree h-version finite elements, Clément's foundational quasi-interpolant uses local averaging on element patches and is defined on L2.Source: §1 · p. 1
- Scott–Zhang quasi-interpolation averages against dual basis functions, making the operator a projection onto the finite-element space.Source: §1 · p. 1
2. Inhomogeneous Boundary Conditions
The section gives an abstract route for incorporating inhomogeneous Dirichlet data into methods designed for homogeneous boundary conditions. Under its stated assumptions, the resulting method is quasi-optimal, with an error bound that accounts for boundary-trace mismatch and approximation.
2. Inhomogeneous Boundary Conditions
- The abstract construction combines a homogeneous-boundary method with a trace projection and an extension into the discrete space; the resulting operator is independent of the chosen extension.Source: §2 · p. 3
- The homogeneous-boundary method is assumed to define a uniformly bounded projection onto the discrete zero-trace space, and the trace map is assumed surjective.Source: §2 · p. 3
- The section defines quasi-optimality of the resulting operator by bounding its error by the best approximation in the full discrete space.Source: §2 · p. 3
- The homogeneous-boundary operator satisfies a quasi-optimal best-approximation bound, with a factor depending on its operator norm.Source: §2 · p. 3
- The inhomogeneous-boundary error is bounded by a term involving the mismatch between the trace and its discrete projection, plus best approximation among discrete functions with the projected trace.Source: §2 · p. 4
3. Space-time Scott–Zhang interpolation operators
The section develops a Scott–Zhang-type space-time projection on tensor-product meshes and identifies a trace-space mapping as a candidate for approximating Dirichlet boundary conditions. The projection preserves representable lateral boundary traces and satisfies local stability estimates, supporting a quasi-optimal space-time FOSLS method for inhomogeneous Dirichlet data.
3.1. Mathematical setting and main result.
- Theorem 2 gives a linear projection onto the tensor-product finite element space that reproduces representable lateral boundary traces and satisfies local stability estimates.The stability constants depend on local quasi-uniformity of the time mesh, shape regularity or local quasi-uniformity of the spatial mesh, and the polynomial degrees.Source: §3.1 · p. 6
- Under parabolic scaling ht ≃ h_x^2, the interpolation estimates yield global stability in the parabolic energy norm.The theorem also notes robust stability for uniform meshes when h_x ≃ ht.Source: §3.1 · p. 6
3.2. Application to inhomogeneous boundary conditions in space-time discretizations.
- For inhomogeneous Dirichlet FOSLS discretizations, the uniformly bounded, boundary-preserving projection Jh yields a quasi-optimal operator Gh under parabolic scaling.The construction combines the space-time interpolant with projections for the other components of the discrete space.Source: §3.2 · p. 8
3.3. Isotropic Scott–Zhang operators.
- The isotropic construction uses modified dual bases supported on Bernstein basis patches, with biorthogonality and bounds controlled by polynomial degree and mesh regularity.These bases provide the ingredients for the spatial and temporal Scott–Zhang operators.Source: §3.3 · pp. 8–9
3.4. Anisotropic Scott–Zhang operators.
- Combining temporal and spatial Scott–Zhang operators gives space-time projections with local stability, measured over element patches containing the dual-basis supports.Pointwise interpolation in time can reduce the temporal operator’s domain of influence.Source: §3.4 · p. 10
3.5. Scott–Zhang-type space-time interpolation operator.
- The final interpolant uses temporal dual bases and spatial boundary-trace dual bases, and is well-defined because energy-space functions have H^1/2,1/4 lateral traces.The construction uses traces of the spatial Bernstein basis on the boundary mesh.Source: §3.5 · pp. 11–12
- The mapping from H1/2,1/4(Σ) into ran trΣ|Sk,ℓ(Tt,x) is a valid candidate for the operator Ph.It is proposed for approximating Dirichlet boundary conditions.Source: §3.5 · p. 15