Donald L. Brown

Donald L. Brown, Dietmar Gallistl, Daniel Peterseim

This paper presents a multiscale Petrov-Galerkin finite element method for time-harmonic acoustic scattering problems with heterogeneous coefficients in the high-frequency regime. We show that the method is pollution- free also in the case of heterogeneous media provided that the stability bound of the continuous problem grows at most polynomially with the wave number k. By generalizing classical estimates of [Melenk, Ph.D. Thesis 1995] and [Hetmaniuk, Commun. Math. Sci. 5 (2007)] for homogeneous medium, we show that this assumption of polynomially wave number growth holds true for a particular class of smooth heterogeneous material coefficients. Further, we present numerical examples to verify our stability estimates and implement an example in the wider class of discontinuous coefficients to show computational applicability beyond our limited class of coefficients.

Donald L. Brown, Joscha Gedicke

In this paper, we develop a numerical multiscale method to solve elliptic boundary value problems with heterogeneous diffusion coefficients and with singular source terms. When the diffusion coefficient is heterogeneous, this adds to the computational costs, and this difficulty is compounded by a singular source term. For singular source terms, the solution does not belong to the Sobolev space $H^1$, but to the space $W^{1,p}$ for some $p<2$. Hence, the problem may be reformulated in a distance-weighted Sobolev space. Using this formulation, we develop a method to upscale the multiscale coefficient near the singular sources by incorporating corrections into the coarse-grid. Using a sub-grid correction method, we correct the basis functions in a distance-weighted Sobolev space and show that these corrections can be truncated to design a computationally efficient scheme with optimal convergence rates. Due to the nature of the formulation in weighted spaces, the variational form must be posed on the cross product of complementary spaces. Thus, two such sub-grid corrections must be computed, one for each multiscale space of the cross product. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Therefore, we develop a weighted projective quasi-interpolation that can be used for a general class of Muckenhoupt weight functions. We verify the optimal convergence of the method in some numerical examples with singular point sources and line fractures, and with oscillatory and heterogeneous diffusion coefficients.

Donald L. Brown, Joscha Gedicke, Daniel Peterseim

In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional Laplacian is a nonlocal operator in its standard form, however the Caffarelli-Silvestre extension allows for a localization of the equations. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a sub-grid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Since the solution of the extended problem on the critical boundary is of main interest, we construct a projective quasi-interpolation that has both $d$ and $d+1$ dimensional averages over subsets in the spirit of the Scott-Zhang operator. We show that this operator satisfies local stability and local approximation properties in weighted Sobolev spaces. We further show that we can obtain a greater rate of convergence for sufficient smooth forces, and utilizing a global $L^2$ projection on the critical boundary. We present some numerical examples, utilizing our projective quasi-interpolation in dimension $2+1$ for analytic and heterogeneous cases to demonstrate the rates and effectiveness of the method.

Donald L. Brown, Daniel Peterseim

In this paper we develop a multiscale method to solve problems in complicated porous microstructures with Neumann boundary conditions. By using a coarse-grid quasi-interpolation operator to define a fine detail space and local orthogonal decomposition, we construct multiscale corrections to coarse-grid basis functions with microstructure. By truncating the corrector functions we are able to make a computationally efficient scheme. Error results and analysis are presented. A key component of this analysis is the investigation of the Poincaré constants in perforated domains as they may contain micro-structural information. Using a constructive method originally developed for weighted Poincaré inequalities, we are able to obtain estimates on Poincaré constants with respect to scale and separation length of the pores. Finally, two numerical examples are presented to verify our estimates.