We present and analyze a methodology for numerical homogenization of spatial networks models, e.g. heat conduction and linear deformation in large networks of slender objects, such as paper fibers. The aim is to construct a coarse model of the problem that maintains high accuracy also on the micro-scale. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data, at a low computational cost. We prove optimal convergence to the micro-scale solution of the proposed method under mild assumptions on the homogeneity, connectivity, and locality of the network on the coarse scale. The theoretical findings are numerically confirmed for both scalar-valued (heat conduction) and vector-valued (linear deformation) models.

Mathematics, Matematik, numerical homogenization, localized orthogonal decomposition, stochastic partial differential equations, (multilevel) Monte Carlo methods, strong convergence, and multiscale method

Abstract

A multiscale method is proposed for a parabolic stochastic partial differential equation with additive noise and highly oscillatory diffusion. The framework is based on the localized orthogonal decomposition (LOD) method and computes a coarse -scale representation of the elliptic operator, enriched by fine -scale information on the diffusion. Optimal order strong convergence is derived. The LOD technique is combined with a (multilevel) Monte Carlo estimator and the weak error is analyzed. Numerical examples that confirm the theoretical findings are provided, and the computational efficiency of the method is highlighted.

Computer Methods in Applied Mechanics and Engineering. 410

Subjects

Wave equation, Localized orthogonal decomposition, Network model, and Numerical homogenization

Abstract

We present and analyze a multiscale method for wave propagation problems, posed on spatial networks. By introducing a coarse scale, using a finite element space interpolated onto the network, we construct a discrete multiscale space using the localized orthogonal decomposition (LOD) methodology. The spatial discretization is then combined with an energy conserving temporal scheme to form the proposed method. Under the assumption of well-prepared initial data, we derive an a priori error bound of optimal order with respect to the space and time discretization. In the analysis, we combine the theory derived for stationary elliptic problems on spatial networks with classical finite element results for hyperbolic problems. Finally, we present numerical experiments that confirm our theoretical findings.

In this thesis we develop and analyze generalized finite element methods for time-dependent partial differential equations (PDEs). The focus lies on equations with rapidly varying coefficients, for which the classical finite element method is insufficient, as it requires a mesh fine enough to resolve the data. The framework for the novel methods are based on the localized orthogonal decomposition (LOD) technique. The main idea of this method is to construct a modified finite element space whose basis functions contain information about the variations in the coefficients, hence yielding better approximation properties. At first, the localized orthogonal decomposition framework is extended to the strongly damped wave equation, where two different highly varying coefficients are present (Paper~I). The dependency of the solution on the different coefficients vary with time, which the proposed method accounts for automatically. Then we consider a parabolic equation where the diffusion is rapidly varying in both time and space (Paper~II). Here, the framework is extended so that the modified finite element space uses space-time basis functions that contain the information of the diffusion coefficient. Furthermore, we study wave propagation problems posed on spatial networks (Paper~III). Such systems are characterized by a matrix with large variations inherited from the underlying network. For this purpose, an LOD based approach adapted to general matrix systems is considered. Finally, we analyze the framework for a parabolic stochastic PDE with multiscale characteristics (Paper~IV). In all papers we prove error estimates for the methods, and confirm the theoretical findings with numerical examples.

Hellman, Fredrik, Malqvist, Axel, and Wang, Siyang

Mathematical Modelling and Numerical Analysis. 55:S761-S784

Subjects

Generalized finite element method, localized orthogonal decomposition, porous media, fracture, and Darcy flow

Abstract

We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An a priori error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems.

Mathematical Modelling and Numerical Analysis. 55(4):1375-1403

Subjects

Strongly damped wave equation, Reduced basis method, Localized orthogonal decomposition, Finite element method, and Multiscale

Abstract

We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced in Malqvist and Peterseim [Math. Comp. 83 (2014) 2583-2603], and is designed to handle independent variations in both the damping and the wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal order is proven in L2(H1)-norm, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.

Gross–Pitaevskii equation, finite elements, Bose–Einstein condensates, ground states, localized orthogonal decomposition, Beräkningsvetenskap med inriktning mot numerisk analys, and Scientific Computing with specialization in Numerical Analysis

Abstract

Abstract. In this paper we revisit a two-level discretization based on localized orthogonal decomposition (LOD). It was originally proposed in [P. Henning, A. Målqvist, and D. Peterseim, SIAM J. Numer. Anal., 52 (2014), pp. 1525-1550] to compute ground states of Bose-Einstein condensates by finding discrete minimizers of the Gross-Pitaevskii energy functional. The established convergence rates for the method appeared, however, suboptimal compared to numerical observations and a proof of optimal rates in this setting remained open. In this paper we shall close this gap by proving optimal order error estimates for the \(L^2\) - and \(H^1\) -errors between the exact ground state and discrete minimizers, as well as error estimates for the ground state energy and the ground state eigenvalue. In particular, the achieved convergence rates for the energy and the eigenvalue are of 6th order with respect to the mesh size on which the discrete LOD space is based, without making any additional regularity assumptions. These high rates justify the use of very coarse meshes, which significantly reduces the computational effort for finding accurate approximations of ground states. In addition, we include numerical experiments that confirm the optimality of the new theoretical convergence rates, for both smooth and discontinuous potentials.

Målqvist, Axel, Persson, Anna, and Stillfjord, Tony

SIAM Journal of Scientific Computing. 40(4):A2406-A2426

Subjects

Differential Riccati equations, Multiscale, Linear quadratic regulator problems, Finite elements, and Localized orthogonal decomposition

Abstract

We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic problems, standard finite element discretizations perform poorly in this setting unless the grid resolves the fine-scale features of the problem. This results in unfeasible amounts of computation and high memory requirements. In this paper, we demonstrate how the localized orthogonal decomposition method may be used to acquire accurate results also for coarse discretizations, at the low cost of solving a series of small, localized elliptic problems. We prove second-order convergence (except for a logarithmic factor) in the L2operator norm and first-order convergence in the corresponding energy norm. These results are both independent of the multiscale variations in the state equation. In addition, we provide a detailed derivation of the fully discrete matrix-valued equations and show how they can be handled in a low-rank setting for large-scale computations. In connection to this, we also show how to efficiently compute the relevant operator-norm errors. Finally, our theoretical results are validated by several numerical experiments.

localized orthogonal decomposition, multiscale, Strongly damped wave equation, parabolic equations., and finite element method

Abstract

In this thesis we develop and analyze generalized finite element methods for time-dependent partial differential equations (PDEs). The focus lies on equa- tions with rapidly varying coefficients, for which the classical finite element method is insufficient, as it requires a mesh fine enough to resolve the data. The framework for the novel methods are based on the localized orthogonal decomposition technique. The main idea of this method is to construct a modified finite element space whose basis functions contain information about the variations in the coefficients, hence yielding better ap- proximation properties. At first, the localized orthogonal decomposition framework is extended to the strongly damped wave equation, where two different highly varying coeffi- cients are present (Paper I). The dependency of the solution on the different coefficients vary with time, which the proposed method accounts for automat- ically. Then we consider a parabolic equation where the diffusion is rapidly varying in both time and space (Paper II). Here, the framework is extended so that the modified finite element space uses space-time basis functions that contain the information of the diffusion coefficient. In both papers we prove error estimates for the methods, and confirm the theoretical findings with numerical examples.

regularity, generalized finite element, localized orthogonal decomposition, multiscale, finite element method, Thermoelasticity, Riccati equations, thermistor, linear elasticity, Joule heating, and parabolic equations

Abstract

In this thesis we study numerical methods for evolution problems in multiphysics. The term multiphysics is commonly used to describe physical phenomena that involve several interacting models. Typically, such problems result in coupled systems of partial differential equations. This thesis is essentially divided into two parts, which address two different topics with applications in multiphysics. The first topic is numerical analysis for multiscale problems, with a particular focus on heterogeneous materials, like composites. For classical finite element methods such problems are known to be numerically challenging, due to the rapid variations in the data. One of our main goals is to develop a numerical method for the thermoelastic system with multiscale coefficients. The method we propose is based on the localized orthogonal decomposition (LOD) technique introduced in Målqvist and Peterseim (Math Comput 83(290):2583-2603, 2014). This is performed in three steps, first we extend the LOD framework to parabolic problems (Paper I) and then to linear elasticity equations (Paper II). Using the theory developed in these two papers we address the thermoelastic system (Paper III). In addition, we aim to extend the LOD framework to differential Riccati equations where the state equation is governed by a multiscale operator. The numerical solution of such problems involves solving many parabolic equations with multiscale coefficients. Hence, by applying the method developed in Paper I to Riccati equations the computational gain may be significantly large. In this thesis we show that this is indeed the case (Paper IV). The second part of this thesis is devoted to the Joule heating problem, a coupled nonlinear system describing the temperature and electric current in a material. Analyzing this system turns out to be difficult due to the low regularity of the nonlinear term. We overcome this issue by introducing a new variational formulation based on a cut-off functional. Using this formulation, we prove (Paper V) strong convergence of a large class of finite element methods for the Joule heating problem with mixed boundary conditions on nonsmooth domains in three dimensions.

In this paper we consider a numerical homogenization technique for curl-curl problems that is based on the framework of the localized orthogonal decomposition and which was proposed in [D. Callisti, P. Henning, and B. Verfürth, SIAM J. Numer. Anal, 56 (2018), pp. 1570-1596] for problems with essential boundary conditions. The findings of the aforementioned work establish quantitative homogenization results for the time-harmonic Maxwell's equations that hold beyond assumptions of periodicity; however, a practical realization of the approach was left open. In this paper, we transfer the findings from essential boundary conditions to natural boundary conditions, and we demonstrate that the approach yields a computable numerical method. We also investigate how boundary values of the source term can affect the computational complexity and accuracy. Our findings will be supported by various numerical experiments, both in 2D and 3D.

Engwer, Christian, Henning, Patrick, Målqvist, Axel, and Peterseim, Daniel

Computer Methods in Applied Mechanics and Engineering. 350:123-153

Subjects

Mathematics, Matematik, Efficient numerical solvers, Linear solvers, Localized orthogonal decomposition, Multiscale finite elements, Multiscale methods, and Subscale correction methods

Abstract

In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems.