A priori error estimates of regularized elliptic problems

Abstract

Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $$H^1$$ H 1 and $$L^2$$ L 2 error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.

Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

“…Since the hypersingular source term is not in , we resort to a regularized strategy 10 which allows one to replace the Dirac delta distribution of the source terms in ( 11 ) and ( 12 ) with a smooth approximation with compact support of radius . The analysis in 10 suggests that the support of the approximated Dirac distribution should be chosen proportional to the grid size h , in order to obtain the best results in terms of convergence rates and computational costs.…”

“…With the purpose of enhancing the quality and the outreach of MRE analysis, this paper addresses the issues of developing a computational multiscale model that can account for an arbitrary complexity of the (microscale) fluid vasculature and, at the same time, be efficiently upscaled for the application in the context of (coarse) MRE data. To this aim, we extend the immersed multiscale framework recently proposed in, 9 , 10 based on describing the tissue as an elastic material, taking into account the presence of the fluid network as a singular forcing term. In particular, we address the coupling between the three-dimensional elastic matrix and a finite volume one-dimensional blood flow model that can efficiently handle complex vasculature networks.…”

Adaptive finite element approximations for elliptic problems using regularized forcing data

“…That is, we replace δ with a family of Dirac delta approximations δ r , where r denotes the regularization parameter so that the regularized data, denoted by F r , satisfies certain smoothness property. The error between the exact solution u and its regularized counterpart u r is analyzed in [26] in both the H 1 and L 2 sense. The finite element approximation of (1.1) using quasi-uniform subdivisions is also discussed in [26].…”

“…The error between the exact solution u and its regularized counterpart u r is analyzed in [26] in both the H 1 and L 2 sense. The finite element approximation of (1.1) using quasi-uniform subdivisions is also discussed in [26].…”

“…The approximation error based on regularized data consists of two parts: the regularization error for u and the finite element approximation error for u r . The analysis of adaptive algorithms applied to the regularized problem is complicated by the fact that optimal choices of the regularization parameter r depend on the local mesh size h (see [26]), and that the error estimates depend both on the local mesh size and on the regularization parameter r.…”