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In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.

In this paper, we describe an efficient approach for quantifying uncertainty in two-phase flow applications due to perturbations of the permeability in a multiscale heterogeneous porous medium. The method is based on the application of the multiscale finite element method within the framework of Monte Carlo simulation and an efficient preprocessing construction of the multiscale basis functions. The quantities of interest for our applications are the Darcy velocity and breakthrough time and we quantify their uncertainty by constructing the respective cumulative distribution functions. For the Darcy velocity we use the multiscale finite element method, but due to lack of conservation, we apply the multiscale finite volume element method as an alternative for use with the two-phase flow problem. We provide a number of numerical examples to illustrate the performance of the method.

Nonparametric density estimation for randomly perturbed elliptic problems II : Applications and adaptive modeling

We present a mixed adaptive variational multiscale method for solving elliptic second-order problems. This work is an extension of the adaptive variational multiscale method (AVMS), introduced by Larson and Målqvist,15–17 to a mixed formulation. The method is based on a particular splitting into coarse and fine scales together with a systematic technique for approximation of the fine scale part based on solution of decoupled localized subgrid problems. We present the mixed AVMS method and derive a posteriori error estimates both for linear functionals and the energy norm. Based on the estimates we propose adaptive algorithms for automatic tuning of critical discretization parameters. Finally, we present numerical examples on a two-dimensional slice of an oil reservoir.

The adaptive variational multiscale method is an extension of the variational multiscale method where the line-scale part of the solution is approximated by a sum of numerically computed solutions ...

In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart-Thomas finite element spaces. It requires to solve local problems in small patches around the elements of a coarse grid. These computations can be perfectly parallelized and are cheap to perform. Using the results of these patch problems, we construct a low dimensional multiscale mixed finite element space with very high approximation properties. This space can be used for solving the original saddle point problem in an efficient way. We prove convergence of our approach, independent of structural assumptions or scale separation. Finally, we demonstrate the applicability of our method by presenting a variety of numerical experiments, including a comparison with an MsFEM approach.

The accuracy of many multiscale methods based on localized computations suffers from high contrast coefficients since the localization error generally depends on the contrast. We study a class of methods based on the variational multiscale method, where the range and kernel of a quasi-interpolation operator de fines the method. We present a novel interpolation operator for two-valued coefficients and prove that it yields contrast independent localization error under physically justified assumptions on the geometry of inclusions and channel structures in the coefficient. The idea developed in the paper can be transferred to more general operators and our numerical experiments show that the contrast independent localization property follows.

Goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations

We develop a new adaptive multiscale finite element method using the variational multiscale framework together with a systematic technique for approximation of the fine scale part of the solution. The fine scale is approximated by a sum of solutions to decoupled localized problems, which are solved numerically on a fine grid partition of a patch of coarse grid elements. The sizes of the patches of elements may be increased to control the error caused by localization. We derive an a posteriori error estimate in the energy norm which captures the dependency of the crucial discretization parameters: the coarse grid mesh size, the fine grid mesh size, and the sizes of the patches. Based on the a posteriori error estimate we present an adaptive algorithm that automatically tunes these parameters. Finally, we show how the method works in practice by presenting various numerical examples.

We use the local orthogonal decomposition technique introduced in Målqvist and Peterseim (Math Comput 83(290):2583–2603, 2014) to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale coefficients. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations of the coefficients, is proven in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\infty (L_2)$$\end{document}L∞(L2)-norm. We present numerical examples, which confirm our theoretical findings.

We consider the time-dependent Gross-Pitaevskii equation describing the dynamics of rotating Bose-Einstein condensates and its discretization with the finite element method. We analyze a mass conserving Crank-Nicolson-type discretization and prove corresponding a priori error estimates with respect to the maximum norm in time and the L2- and energy-norm in space. The estimates show that we obtain optimal convergence rates under the assumption of additional regularity for the solution to the Gross-Pitaevskii equation. We demonstrate the performance of the method in numerical experiments. © by SIAM 2017.

We consider a large-scale quadratic eigenvalue problem (QEP), formulated using P1 finite elements on a fine scale reference mesh. This model describes damped vibrations in a structural mechanical system. In particular we focus on problems with rapid material data variation, e.g., composite materials. We construct a low dimensional generalized finite element (GFE) space based on the localized orthogonal decomposition (LOD) technique. The construction involves the (parallel) solution of independent localized linear Poisson-type problems. The GFE space is then used to compress the large-scale algebraic QEP to a much smaller one with a similar modeling accuracy. The small scale QEP can then be solved by standard techniques at a significantly reduced computational cost. We prove convergence with rate for the proposed method and numerical experiments confirm our theoretical findings.

We propose and analyze a method for computing failure probabilities of systems modeled as numerical deterministic models (e.g., PDEs) with uncertain input data. A failure occurs when a functional of the solution to the model is below (or above) some critical value. By combining recent results on quantile estimation and the multilevel Monte Carlo method we develop a method which reduces computational cost without loss of accuracy. We show how the computational cost of the method relates to error tolerance of the failure probability. For a wide and common class of problems, the computational cost is asymptotically proportional to solving a single accurate realization of the numerical model, i.e., independent of the number of samples. Significant reductions in computational cost are also observed in numerical experiments.

Finite element multiscale methods for Poisson's equation with rapidly varying heterogeneous coefficients

We prove strong convergence of conforming finite element approximations to the stationary Joule heating problem with mixed boundary conditions on Lipschitz domains in three spatial dimensions. We show optimal global regularity estimates on creased domains and prove a priori and a posteriori bounds for shape regular meshes., Comment: Keywords: Joule heating problem, thermistors, a posteriori error analysis, a priori error analysis, finite element method

This work presents a new methodology for computing ground states of Bose-Einstein condensates based on finite element discretizations on two different scales of numerical resolution. In a pre-processing step, a low-dimensional (coarse) generalized finite element space is constructed. It is based on a local orthogonal decomposition and exhibits high approximation properties. The non-linear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low-dimensional space, without loss of accuracy when compared with the solution of the full fine scale problem. The pre-processing step is independent of the types and numbers of bosons. A post-processing step further improves the accuracy of the method. We present rigorous a priori error estimates that predict convergence rates H^3 for the ground state eigenfunction and H^4 for the corresponding eigenvalue without pre-asymptotic effects; H being the coarse scale discretization parameter. Numerical experiments indicate that these high rates may still be pessimistic., Accepted for publication in SIAM J. Numer. Anal., 2014

In this paper we propose a Local Orthogonal Decomposition method (LOD) for elliptic partial differential equations with inhomogeneous Dirichlet- and Neumann boundary conditions. For this purpose, we present new boundary correctors which preserve the common convergence rates of the LOD, even if the boundary condition has a rapidly oscillating fine scale structure. We prove a corresponding a-priori error estimate and present numerical experiments. We also demonstrate numerically that the method is reliable with respect to thin conductivity channels in the diffusion matrix. Accurate results are obtained without resolving these channels by the coarse grid and without using patches that contain the channels.

We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we consider benchmark multi-scale eigenvalue problems in reservoir modeling and material science. We compute a low-dimensional generalized (possibly mesh free) finite element space that preserves the lowermost eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of $H^1_0(\Omega)$ by means of a certain Cl\'ement-type quasi-interpolation operator., Comment: to appear in Numerische Mathematik

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.

In this paper we present the adaptive variational multiscale method for solving the Poisson equation in mixed form. We use the method introduced in Larson and Malqvist (Comput Method Appl Mech Eng 196:2313–2324, 2007), and further analyzed and applied to mixed problems in Larson and Malqvist (Comput Method Appl Mech Eng 19:1017–1042, 2009), which is a general tool for solving linear partial differential equations with multiscale features in the coefficients. We extend the numerics in Larson and Malqvist (Comput Method Appl Mech Eng 19:1017–1042, 2009) from rectangular meshes to triangular meshes which allow for computation on more complicated domains. A new a posteriori error estimate is also included, which is used in an adaptive algorithm. We present a numerical example that shows the efficiency of incorporating a posteriori based adaptivity into the method.

This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size H H , patches of diameter H log ⁡ ( 1 / H ) H\log (1/H) are sufficient to preserve a linear rate of convergence in H H without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods.

The variational multiscale method (VMM) provides a general framework for construction of multiscale finite element methods. In this paper we propose a method for parallel solution of the fine scale problem based on localized Dirichlet problems which are solved numerically. Next we present a posteriori error representation formulas for VMM which relates the error in linear functionals to the discretization errors, resolution and size of patches in the localized problems, in the fine scale approximation. These formulas are derived by using duality techniques. Based on the a posteriori error representation formula we propose an adaptive VMM with automatic tuning of the critical parameters. We primarily study elliptic second order partial differential equations with highly oscillating coefficients or localized singularities.

Numerical analysis and adaptive computation for solutions of elliptic problems with randomly perturbed coefficients

We consider a system of equations that model the temperature, electric potential and deformation of a thermoviscoelastic body. A typical application is a thermistor; an electrical component that can be used e.g. as a surge protector, temperature sensor or for very precise positioning. We introduce a full discretization based on standard finite elements in space and a semi-implicit Euler-type method in time. For this method we prove optimal convergence orders, i.e. second-order in space and first-order in time. The theoretical results are verified by several numerical experiments in two and three dimensions., Comment: 20 pages, 6 figures, 2 tables

Many applications are tackled using the Laplace Transform (LT) known on a countable number of real values [J. Electroanal. Chem. 608, 37–46 (2007), Int. J. solid Struct. 41, 3653–3674 (2004), Imaging 26, 1183–1196 (2008), J. Magn. Reson. 156, 213–221 (2002)]. The usual approach to solve the LT inverse problem relies on a regularization technique combined with information a priori both on the LT function and on its inverse (see for instance [http://s-provencher.com/pages/ contin.shtml]). We propose a fitting model enjoying LT properties: we define a generalized spline that interpolates the LT function values and mimics the asymptotic behavior of LT functions. Then, we prove existence and uniqueness of this model and, through a suitable error analysis, we give a priori approximation error bounds to confirm the reliability of this approach. Numerical results are presented.