Optimal Finite-Element Interpolation on Curved Domains

Abstract

We prove pointwise a posteriori error estimates for semi-and fullydiscrete finite element methods for approximating the solution u to a parabolic model problem. Our estimates may be used to bound the finite element error u − u h L ∞ (D) , where D is an arbitrary subset of the space-time domain of the definition of the given PDE. In contrast to standard global error estimates, these estimators de-emphasize spatial error contributions from space-time regions removed from D. Our results are valid on arbitrary shape-regular simplicial meshes which may change in time, and also provide insight into the contribution of mesh change to local errors. When implemented in an adaptive method, these estimates require only enough spatial mesh refinement away from D in order to ensure that local solution quality is not polluted by global effects.

Optimal error Estimates for the Stokes and Navier–Stokes equations with slip–boundary condition

“…1) is a homeomorphism betweenΩ h and Ω. The construction in [4,13] also implies that ΦT = 0 ifT has at most one vertex on ∂Ω h , so that G h ≡ I on all simplices which are disjoint from ∂Ω h . Finally, we have the estimates…”

“…The existence of such a triangulation together with the associated interpolation estimates is proved in [4,13]. In essence, for everyT ∈T h there is a mapping ΦT ∈ C 3 (T ; R d ) such that FT :=FT + ΦT mapsT onto a curved d-simplex T ⊆ Ω and…”

Sharply local pointwise a posteriori error estimates for parabolic problems

“…The first is to employ parametric finite element spaces along with an appropriately defined interpolant. Such spaces and a corresponding interpolant I h of Clément type were constructed for both two-and three-dimensional domains with smooth boundary in [Ber89]. For this interpolant, (2.4) and (2.5) may be trivially obtained from Theorem 4.1 of [Ber89] using the triangle inequality.…”

“…Such spaces and a corresponding interpolant I h of Clément type were constructed for both two-and three-dimensional domains with smooth boundary in [Ber89]. For this interpolant, (2.4) and (2.5) may be trivially obtained from Theorem 4.1 of [Ber89] using the triangle inequality. The same theorem contains (2.6) and (2.7), but with full norms on the right-hand side instead of seminorms.…”

Extension of Some Positive and Linear Operators to Domains with Curved Sides

“…In order to match all the boundary information on a curved domain (as in Dirichlet, Neumann or Robin boundary conditions for differential equation problems), there were considered interpolation operators on domains with curved sides (see, e.g., [4], [6], [9]- [12], [15], [16], [20], [21], [23]). Approximation operators on polygonal domains with some curved sides have also important applications especially in finite element method for differential equations with given boundary conditions and in the piecewise generation of surfaces in computer aided geometric design.…”