Anna Persson

Patrick Henning, Anna Persson

We propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients. The method is formulated in the framework of localized orthogonal decomposition techniques introduced by Målqvist and Peterseim (Math. Comp., 83(290): 2583--2603, 2014). Assuming only $L_\infty$-coefficients we prove linear convergence in the $H^1$-norm, also for materials with large Lamé parameter $λ$. The theoretical a priori error estimate is confirmed by numerical examples.

Axel Målqvist, Anna Persson, Tony Stillfjord

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 $L^2$ operator 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.

Axel Målqvist, Anna Persson

We propose and analyze a generalized finite element method designed for linear quasistatic thermoelastic systems with spatial multiscale coefficients. The method is based on the local orthogonal decomposition technique introduced by Målqvist and Peterseim (Math. Comp., 83(290): 2583--2603, 2014). We prove convergence of optimal order, independent of the derivatives of the coefficients, in the spatial $H^1$-norm. The theoretical results are confirmed by numerical examples.

Max Jensen, Axel Målqvist, Anna Persson

We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on creased domains and additional regularity in the interior of the domain. Due to a variational formulation with a cut-off functional the convergence analysis does not require a discrete maximum principle, permitting approximation spaces suitable for adaptive mesh refinement, responding to the the difference in regularity within the domain.

Mats G. Larson, Carl Lundholm, Anna Persson

We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution operator by a multilayer perceptron (MLP) that takes problem data variables as input and gives a prediction of the finite element solution as output. The variables will typically correspond to parameters in a parametrization of input data such as boundary conditions, coefficients, and right-hand sides. The output will be an approximation of the corresponding finite element solution, thus enabling support and enhancement by the standard finite element method (FEM) both theoretically and practically. The loss function is most often an energy functional and we formulate efficient parallelizable training algorithms based on assembling the energy locally on each element. For large problems, the learning process can be made more efficient by using only a small fraction of randomly chosen elements in the mesh in each iteration. The approach is evaluated on several relevant test cases, where learning the finite element solution operator turns out to be beneficial, both in its own right but also by combination with standard FEM theory and software.

Axel Målqvist, Anna Persson

We use the local orthogonal decomposition technique to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale diffusion coefficient. 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 in the diffusion coefficient, is proven in the $L_\infty(L_2)$-norm. We present numerical examples, which confirm our theoretical findings.