2017/06/09 by Dietmar Gallistl, Gallistl, Dietmar, Patrick Henning +3 · 2 citations
Computer Science · Engineering · #35Q61 #65N12 #65N15 #65N30 #78M10 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Composite Material Mechanics #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1706.02966
openalex publication_date 2017/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
If an elliptic differential operator associated with an H(curl)-problem involves rough (rapidly varying) coefficients, then solutions to the corresponding H(curl)-problem admit typically very low regularity, which leads to arbitrarily bad convergence rates for conventional numerical schemes. The goal of this paper is to show that the missing regularity can be compensated through a corrector operator. More precisely, we consider the lowest order Nédélec finite element space and show the existence of a linear corrector operator with four central properties: it is computable, H(curl)-stable, quasi-local and allows for a correction of coarse finite element functions so that first-order estimates (in terms of the coarse mesh-size) in the H(curl) norm are obtained provided the right-hand side belongs to H(div). With these four properties, a practical application is to construct generalized finite element spaces which can be straightforwardly used in a Galerkin method. In particular, this characterizes a homogenized solution and a first order corrector, including corresponding quantitative error estimates without the requirement of scale separation.