Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

Abstract

Abstract. We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.

Analysis of Finite Element Domain Embedding Methods for Curved Domains using Uniform Grids

“…And the regularity of the solution and the alignment of the domain boundary with the uniform mesh required in this analysis are often not compatible. Furthermore, the results of [17] are actually not valid for general curved domains for Dirichlet problems. In [14,15], an analysis for Neumann problem can be found, which did not impose such restriction on the domain.…”

“…However, when ω is convex and Γ aligns with the mesh-line, as assumed in [17], the result (1.4) remains valid, which needs a different argument.…”

“…A first step in these methods is to extend the boundary value problem to a larger rectangle. There are many ways to do this, one of which is using regularization for Neumann or Robin problem, or penalty for Dirichlet problem, or a combination of these for mixed boundary value problem [14,15,17,23]. A more general approach, including some additional treatments on the original domain boundary, can be found in [2].…”

“…In the literature, there are numerous works devoted to this subject. In [17], a thorough analysis is presented under the assumption that the original domain boundary aligns with the uniform mesh. This assumption seems a stringent restriction on the arbitrariness of the domain.…”

The Approximation and Computation of a Basis of the Trace Space H 1/2

“…There exists a vast literature dealing with the solution of elliptic equations on perforated domains. Particular topics include homogenization (see Buttazzo [4], Cioranescu and Murat [7], Jikov et al [10], Knyazev and Windlund [12]; scattering, see Piat and Codegone [15]; spectral methods, see Cao and Cui [5]; and the finite element method, see Carstensen and Sauter [6], Strouboulis et al [17]). The theory of the Steklov eigenpairs developed by Auchmuty and outlined in Sec.…”