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.
In this paper, we develop a space-time upscaling framework that can be used for many challenging porous media applications without scale separation and high contrast. Our main focus is on nonlinear differential equations with multiscale coefficients. The framework is built on nonlinear nonlocal multi-continuum upscaling concept and significantly extends the results in the proceeding paper. Our approach starts with a coarse space-time partition and identifies test functions for each partition, which plays a role of multi-continua. The test functions are defined via optimization and play a cruci…
Nonlocal multicontinuum (NLMC) upscaling of mixed dimensional coupled flow problem for embedded and discrete fracture models
Texas A&M University（德克萨斯农工大学）；North-Eastern Federal University（东北联邦大学）；The Chinese University of Hong Kong (CUHK)（香港中文大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
In this work, we present a novel nonlocal nonlinear coarse grid approximation using a machine learning algorithm. We consider unsaturated and two-phase flow problems in heterogeneous and fractured porous media, where mathematical models are formulated as general multicontinuum models. We construct a fine grid approximation using the finite volume method and embedded discrete fracture model. Macroscopic models for these complex nonlinear systems require nonlocal multicontinua approaches, which are developed in earlier works. These rigorous techniques require complex local computations, which in…
Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in $H^γ$
Hausdorff-Center for Mathematics（豪斯多夫数学中心）；Vietnam National University, Hanoi, Information Technology Institute（越南国家大学河内分校信息技术研究所）；Friedrich-Schiller-University Jena（耶拿弗里德里希·席勒大学）
A number of different multiscale methods have been developed as a robust alternative to upscaling and as a means for accelerated reservoir simulation of high-resolution geomodels. In their basic setup, multiscale methods use a restriction operator to construct a reduced system of flow equations on a coarser grid, and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. The prolongation operator consists of basis functions computed numerically by solving localized flow problems. One can use the resulting multiscale solver both as a CPR-prec…
The \emph{deterministic} sparse grid method, also known as Smolyak's method, is a well-established and widely used tool to tackle multivariate approximation problems, and there is a vast literature on it. Much less is known about \emph{randomized} versions of the sparse grid method. In this paper we analyze randomized sparse grid algorithms, namely randomized sparse grid quadratures for multivariate integration on the $D$-dimensional unit cube $[0,1)^D$. Let $d,s \in \mathbb{N}$ be such that $D=d\cdot s$. The $s$-dimensional building blocks of the sparse grid quadratures are based on stratifie…
This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress $\bmσ=-\nabla^{2}u$ is sought in the Sobolev space $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ simultaneously with the displacement $u$ in $L^{2}(Ω)$. Stemming from the structure of $H(\bm{div},Ω;\mathbb{S})$ conforming elements for the linear elasticity problems proposed by J. Hu and S. Zhang, the $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ conforming finite element spaces are constructed by imposing the normal continuity of $\bm{div}\bmσ$…
Reconstruction and Range Characterization for a Directional Confocal Non-Line-of-Sight Imaging Model
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）；Qiuzhen College, Tsinghua University（邱士栋学院，清华大学）
