91 research outputs found
Nonlinear nonlocal multicontinua upscaling framework and its applications
In this paper, we discuss multiscale methods for nonlinear problems. The main
idea of these approaches is to use local constraints and solve problems in
oversampled regions for constructing macroscopic equations. These techniques
are intended for problems without scale separation and high contrast, which
often occur in applications. For linear problems, the local solutions with
constraints are used as basis functions. This technique is called Constraint
Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM).
GMsFEM identifies macroscopic quantities based on rigorous analysis. In
corresponding upscaling methods, the multiscale basis functions are selected
such that the degrees of freedom have physical meanings, such as averages of
the solution on each continuum.
This paper extends the linear concepts to nonlinear problems, where the local
problems are nonlinear. The main concept consists of: (1) identifying
macroscopic quantities; (2) constructing appropriate oversampled local problems
with coarse-grid constraints; (3) formulating macroscopic equations. We
consider two types of approaches. In the first approach, the solutions of local
problems are used as basis functions (in a linear fashion) to solve nonlinear
problems. This approach is simple to implement; however, it lacks the nonlinear
interpolation, which we present in our second approach. In this approach, the
local solutions are used as a nonlinear forward map from local averages
(constraints) of the solution in oversampling region. This local fine-grid
solution is further used to formulate the coarse-grid problem. Both approaches
are discussed on several examples and applied to single-phase and two-phase
flow problems, which are challenging because of convection-dominated nature of
the concentration equation
Computational Multiscale Methods for Linear Poroelasticity with High Contrast
In this work, we employ the Constraint Energy Minimizing Generalized
Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear
heterogeneous poroelasticity with coefficients of high contrast. The proposed
method makes use of the idea of energy minimization with suitable constraints
in order to generate efficient basis functions for the displacement and the
pressure. These basis functions are constructed by solving a class of local
auxiliary optimization problems based on eigenfunctions containing local
information on the heterogeneity. Techniques of oversampling are adapted to
enhance the computational performance. Convergence of first order is shown and
illustrated by a number of numerical tests.Comment: 14 pages, 9 figure
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