A multiscale method for heterogeneous bulk-surface coupling

In this paper, we construct and analyze a multiscale (finite element) method for parabolic problems with heterogeneous dynamic boundary conditions. As origin, we consider a reformulation of the system in order to decouple the discretization of bulk and surface dynamics. This allows us to combine multiscale methods on the boundary with standard Lagrangian schemes in the interior. We prove convergence and quantify explicit rates for low-regularity solutions, independent of the oscillatory behavior of the heterogeneities. As a result, coarse discretization parameters, which do not resolve the fine scales, can be considered. The theoretical findings are justified by a number of numerical experiments including dynamic boundary conditions with random diffusion coefficients

Fast Numerical Methods for Non-local Operators

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Retarded boundary integral equations on the sphere: exact and numerical solution

In this paper we consider the three-dimensional wave equation in unbounded domains with Dirichlet boundary conditions. We start from a retarded single-layer potential ansatz for the solution of these equations which leads to the retarded potential integral equation on the bounded surface of the scatterer. We formulate an algorithm for the space-time Galerkin discretization with smooth and compactly supported temporal basis functions, which were introduced in Sauter & Veit (2013, Numer. Math., 145-176). For the debugging of an implementation and for systematic parameter tests it is essential to have at hand some explicit representations and some analytic properties of the exact solutions for some special cases. We will derive such explicit representations for the case where the scatterer is the unit ball. The obtained formulas are easy to implement and we will present some numerical experiments for these cases to illustrate the convergence behaviour of the proposed metho

Numerical observers with vanishing viscosity for the 1d wave equation

International audienceWe consider a numerical scheme associated with the iterative method developed in [Ramdani-Tucsnak-Weiss] to recover initial conditions of conservative systems. In this method, the initial conditions are reconstructed by using observers. Here we use a finite-difference discretization in space of these observers and our aim is to prove estimates of the errors with respect to the mesh size and to the number of steps in the iterative method. This is done in the particular example of the 1d wave equation. In order to avoid restrictions of the number of steps with respect to the mesh size, we add a numerical viscosity in the numerical observers. A generalization for other equations is also given

Numerical homogenization: multi-resolution and super-localization approaches

Multi-scale problems arise in many scientific and engineering applications, where the effective behavior of a system is determined by the interaction of effects at multiple scales. To accurately simulate such problems without globally resolving all microscopic features, numerical homogenization techniques have been developed. One such technique is the Localized Orthogonal Decomposition (LOD). It provides reliable approximations at coarse discretization levels using problem-adapted basis functions obtained by solving local sub-scale correction problems. This allows the treatment of problems with heterogeneous coefficients without structural assumptions such as periodicity or scale separation. This thesis presents recent achievements in the field of LOD-based numerical homogenization. As a starting point, we introduce a variant of the LOD and provide a rigorous error analysis. This LOD variant is then extended to the multi-resolution setting using the Helmholtz problem as a model problem. The multi-resolution approach allows to improve the accuracy of an existing LOD approximation by adding more discretization levels. All discretization levels are decoupled, resulting in a block-diagonal coarse system matrix. We provide a wavenumber-explicit error analysis that shows convergence under mild assumptions. The fast numerical solution of the block-diagonal coarse system matrix with a standard iterative solver is demonstrated. We further present a novel LOD-based numerical homogenization method named Super-Localized Orthogonal Decomposition (SLOD). The method identifies basis functions that are significantly more local than those of the LOD, resulting in reduced computational cost for the basis computation and improved sparsity of the coarse system matrix. We provide a rigorous error analysis in which the stability of the basis is quantified a posteriori. However, for challenging problems, basis stability issues may arise degrading the approximation quality of the SLOD. To overcome these issues, we combine the SLOD with a partition of unity approach. The resulting method is conceptually simple and easy to implement. Higher order versions of this method, which achieve higher order convergence rates using only the regularity of the source term, are derived. Finally, a local reduced basis (RB) technique is introduced to address the challenges of parameter-dependent multi-scale problems. This method integrates a RB approach into the SLOD framework, enabling an efficient generation of reliable coarse-scale models of the problem. Due to the unique localization properties of the SLOD, the RB snapshot computation can be performed on particularly small patches, reducing the offline and online complexity of the method. All theoretical results of this thesis are supported by numerical experiments