A localized orthogonal decomposition method for semi-linear elliptic problems

In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations in small patches that have a diameter of order H |log H| where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space

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

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

Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods

In this paper, we discuss a general multiscale model reduction framework based on multiscale finite element methods. We give a brief overview of related multiscale methods. Due to page limitations, the overview focuses on a few related methods and is not intended to be comprehensive. We present a general adaptive multiscale model reduction framework, the Generalized Multiscale Finite Element Method. Besides the method's basic outline, we discuss some important ingredients needed for the method's success. We also discuss several applications. The proposed method allows performing local model reduction in the presence of high contrast and no scale separation

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

Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model

In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are therefore necessary. To handle the resulting nonlinear monotone quasilinear elliptic equation, we use linearization based on the Picard iteration. We consider two types of basis functions, offline and online basis functions, following the general framework of GMsFEM. The offline basis functions depend nonlinearly on the solution. Thus, we design an indicator function and we will recompute the offline basis functions when the indicator function predicts that the material property has significant change during the iterations. On the other hand, we will use the residual based online basis functions to reduce the error substantially when updating basis functions is necessary. Our numerical results show that the above combination of offline and online basis functions is able to give accurate solutions with only a few basis functions per each coarse region and updating basis functions in selected iterations.Comment: 19 pages, 2 figures, submitted to Journal of Computational and Applied Mathematic

A reduced basis localized orthogonal decomposition

In this work we combine the framework of the Reduced Basis method (RB) with the framework of the Localized Orthogonal Decomposition (LOD) in order to solve parametrized elliptic multiscale problems. The idea of the LOD is to split a high dimensional Finite Element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information. The low dimensional space is spanned by locally supported basis functions associated with the node of a coarse mesh obtained by solving decoupled local problems. However, for parameter dependent multiscale problems, the local basis has to be computed repeatedly for each choice of the parameter. To overcome this issue, we propose an RB approach to compute in an "offline" stage LOD for suitable representative parameters. The online solution of the multiscale problems can then be obtained in a coarse space (thanks to the LOD decomposition) and for an arbitrary value of the parameters (thanks to a suitable "interpolation" of the selected RB). The online RB-LOD has a basis with local support and leads to sparse systems. Applications of the strategy to both linear and nonlinear problems are given

Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization

The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities

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