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Free, publicly-accessible full text available January 1, 2023
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We present an optimal a priori error estimates of the elliptic problems with Dirac sources away from the singular point using discontinuous and enriched Galerkin finite element methods. It is widely shown that the finite element solutions for elliptic problems with Dirac source terms converge sub-optimally in classical norms on uniform meshes. However, here we employ inductive estimates and norm to obtain the optimal order by excluding the small ball regions with the singularities for both two and three dimensional domains. Numerical examples are presented to substantiate our theoretical results.
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This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the heterogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures themore »
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In this work, we review and describe our computational framework for solving multiphysics phase-field fracture problems in porous media. Therein, the following five coupled nonlinear physical models are addressed: displacements (geo-mechanics), a phase-field variable to indicate the fracture position, a pressure equation (to describe flow), a proppant concentration equation, and/or a saturation equation for two-phase fracture flow, and finally a finite element crack width problem. The overall coupled problem is solved with a staggered solution approach, known in subsurface modeling as the fixed-stress iteration. A main focus is on physics-based discretizations. Galerkin finite elements are employed for the displacement-phase-field systemmore »
