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

Localized Orthogonal Decomposition for two-scale Helmholtz-type problems

In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivated from the homogenization of the Helmholtz equation with high contrast, studied together with a corresponding multiscale method in (Ohlberger, Verf\"urth. A new Heterogeneous Multiscale Method for the Helmholtz equation with high contrast, arXiv:1605.03400, 2016). There, an unavoidable resolution condition on the mesh sizes in terms of the wave number has been observed, which is known as "pollution effect" in the finite element literature. Following ideas of (Gallistl, Peterseim. Comput. Methods Appl. Mech. Engrg. 295:1-17, 2015), we use standard finite element functions for the trial space, whereas the test functions are enriched by solutions of subscale problems (solved on a finer grid) on local patches. Provided that the oversampling parameter $m$, which indicates the size of the patches, is coupled logarithmically to the wave number, we obtain a quasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thus overcoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in the coupling of the function spaces and in the periodic boundary conditions.Comment: 20 page

An adaptive stabilized finite element method for the generalized Stokes problem

In this work we present an adaptive strategy (based on an a posteriori error estimator) for a stabilized finite element method for the Stokes problem, with and without a reaction term. The hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble-like functions. An equivalence result between the norm of the finite element error and the estimator is given, where the dependence of the constants on the physics of the problem is explicited. Several numerical results confirming both the theoretical results and the good performance of the estimator are given

Numerical homogenization of H(curl)-problems

If an elliptic differential operator associated with an $\mathbf{H}(\mathrm{curl})$-problem involves rough (rapidly varying) coefficients, then solutions to the corresponding $\mathbf{H}(\mathrm{curl})$-problem admit typically very low regularity, which leads to arbitrarily bad convergence rates for conventional numerical schemes. The goal of this paper is to show that the missing regularity can be compensated through a corrector operator. More precisely, we consider the lowest order N\'ed\'elec finite element space and show the existence of a linear corrector operator with four central properties: it is computable, $\mathbf{H}(\mathrm{curl})$-stable, quasi-local and allows for a correction of coarse finite element functions so that first-order estimates (in terms of the coarse mesh-size) in the $\mathbf{H}(\mathrm{curl})$ norm are obtained provided the right-hand side belongs to $\mathbf{H}(\mathrm{div})$. With these four properties, a practical application is to construct generalized finite element spaces which can be straightforwardly used in a Galerkin method. In particular, this characterizes a homogenized solution and a first order corrector, including corresponding quantitative error estimates without the requirement of 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 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

Higher-order finite element methods for the nonlinear Helmholtz equation

In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and (pre-asymptotic) error estimates of the finite element solution under a resolution condition between the wave number $k$, the mesh size $h$ and the polynomial degree p of the form “$k(kh)^p$ sufficiently small” and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in $h$ from the case $p = 1$ in [H. Wu, J. Zou, SIAM J. Numer. Anal. 56(3): 1338-1359, 2018] can be removed for $p \ge 2$. We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data