176 research outputs found
Numerical homogenization methods for parabolic monotone problems
In this paper we review various numerical homogenization methods for monotone parabolic problems with multiple scales. The spatial discretisation is based on finite element methods and the multiscale strategy relies on the heterogeneous multiscale method. The time discretization is performed by several classes of Runge-Kutta methods (strongly A-stable or explicit stabilized methods). We discuss the construction and the analysis of such methods for a range of problems, from linear parabolic problems to nonlinear monotone parabolic problems in the very general Lp(W1,p) setting. We also show that under appropriate assumptions, a computationally attractive linearized method can be constructed for nonlinear problems
Linearized numerical homogenization method for nonlinear monotone parabolic multiscale problems
We introduce and analyze an efficient numerical homogenization method for a class of nonlinear parabolic problems of monotone type in highly oscillatory media. The new scheme avoids costly Newton iterations and is linear at both the macroscopic and the microscopic scales. It can be interpreted as a linearized version of a standard nonlinear homogenization method. We prove the stability of the method and derive optimal a priori error estimates which are fully discrete in time and space. Numerical experiments confirm the error bounds and illustrate the efficiency of the methodfor various nonlinear problems
Waveform Relaxation for the Computational Homogenization of Multiscale Magnetoquasistatic Problems
This paper proposes the application of the waveform relaxation method to the
homogenization of multiscale magnetoquasistatic problems. In the monolithic
heterogeneous multiscale method, the nonlinear macroscale problem is solved
using the Newton--Raphson scheme. The resolution of many mesoscale problems per
Gauss point allows to compute the homogenized constitutive law and its
derivative by finite differences. In the proposed approach, the macroscale
problem and the mesoscale problems are weakly coupled and solved separately
using the finite element method on time intervals for several waveform
relaxation iterations. The exchange of information between both problems is
still carried out using the heterogeneous multiscale method. However, the
partial derivatives can now be evaluated exactly by solving only one mesoscale
problem per Gauss point.Comment: submitted to JC
Numerical homogenization for nonlinear strongly monotone problems
In this work we introduce and analyze a new multiscale method for strongly nonlinear monotone equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local fine-scale problems which is then used in a generalized finite element method. The linearity of the fine-scale problems allows their localization and, moreover, makes the method very efficient to use. The new method gives optimal a priori error estimates up to linearization errors beyond periodicity and scale separation and without assuming higher regularity of the solution. The effect of different linearization strategies is discussed in theory and practice. Several numerical examples including stationary Richards equation confirm the theory and underline the applicability of the method
Multiscale Finite Element Methods for Nonlinear Problems and their Applications
In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities
Numerical homogenization for nonlinear strongly monotone problems
In this work we introduce and analyze a new multiscale method for strongly
nonlinear monotone equations in the spirit of the Localized Orthogonal
Decomposition. A problem-adapted multiscale space is constructed by solving
linear local fine-scale problems which is then used in a generalized finite
element method. The linearity of the fine-scale problems allows their
localization and, moreover, makes the method very efficient to use. The new
method gives optimal a priori error estimates up to linearization errors. The
results neither require structural assumptions on the coefficient such as
periodicity or scale separation nor higher regularity of the solution. The
effect of different linearization strategies is discussed in theory and
practice. Several numerical examples including stationary Richards equation
confirm the theory and underline the applicability of the method
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