The general aims of this project are the numerical analysis and simulation of partial differential equations with multiscale character. A vast number of technological applications is based on the mathematical understanding and numerical simulation of acoustic, elastic, or electromagnetic waves. In many cases, the involved materials exhibit a multiscale structure that cannot be computationally resolved even by modern computer facilities.
We consider two diverse aspects of numerical homogenization. For some advanced models from electromagnetism we develop numerical methods that compute an effective limit in the case of scale separation, see section Numerical multiscale methods for electromagnetic waves. For elliptic and time-harmonic problems we explore multiscale methods beyond scale-separation, see section Numerical homogenization beyond scale separation.
Typically, the global or macroscopic response of the material can theoretically be well represented with a computational model of moderate size that does not resolve all the fine features in the multiscale material. But such upscaled or homogenized descriptions need the knowledge of the effective material behavior and it is the computation of these effective material coefficients that makes the numerical simulation of heterogeneous materials challenging.
In order to deal with the rapidly oscillating material parameters, we consider multiscale methods and thereby aim at a feasible numerical simulation. In general, these methods try to decompose the exact solution into a macroscopic contribution (without oscillations), which can be discretized on a coarse mesh, and a fine-scale contribution.
We consider electromagnetic waves in a Debye medium with coefficients that periodically vary on a fine scale \(\delta\). The corresponding Maxwell system in a bounded domain with permittivity \(\varepsilon^\delta\), permeability \(\mu^\delta\) and susceptibility \(\chi^\delta\) reads \[\begin{equation*} \varepsilon^\delta\partial_t\vec{E}^\delta(t) + \int\limits_0^t\chi^\delta(t-s)\vec{E}^\delta(s)\:\mathrm{d} s-\operatorname{curl} \vec{H}^\delta(t) = -\vec{j}(t), \mu^\delta\partial_t\vec{H}^\delta(t) + \operatorname{curl}\vec{E}^\delta(t) =\vec{0}. \end{equation*}\] As mentioned above we use existing theoretical results which lead to an effective system that represents the macroscopic response of the material. \[\begin{equation*} \varepsilon^\mathrm{eff}\partial_t \vec{E}^\mathrm{eff}(t) + \int\limits_0^t \chi^\mathrm{eff}(t-s) \vec{E}^\mathrm{eff}(s) \:\mathrm{d} s -\operatorname{curl}\vec{H}^\mathrm{eff}(t) = -\vec{j}(t) \mu^\mathrm{eff}\partial_t\vec{H}^\mathrm{eff}(t) + \operatorname{curl}\vec{E}^\mathrm{eff}(t) =\vec{0} \end{equation*}\] In this model the parameters no longer depend on the microscopic scale however they are not known analytically. To be more precise the computation of the effective material parameters involves the solution of elliptic partial differential equations, the so called micro problems, for every point in the domain.
We use the Finite Element Heterogeneous Multiscale Method (FE-HMM) as numerical scheme. Here we use Nédélec's curl-conforming finite elements for the macroscopic equation and observe that we can reduce the number of micro problems that have to be solved to the finite number of macroscopic quadrature points. For the discretization of the micro problems we choose standard Lagrange finite elements and since these problems are defined on domains of size \(\delta\) we can resolve the micro scale. Due to the fact that the micro problems are decoupled, they can be solved in parallel. The general procedure of the HMM is shown in Figure 1.
This part of our project is devoted to the question how far such considerations can be extended beyond the (locally) periodic case. It aims at numerical homogenization of \(H(\operatorname{curl})\)-elliptic problems beyond the periodic case and without assuming scale separation. We consider stationary \(H(\operatorname{curl})\)-elliptic problems and establish a decomposition into a coarse and fine part, using a suitable interpolation operator from the Finite Element Exterior Calculus framework. The coarse part gives an optimal approximation in a negative Sobolev norm, and in order to obtain optimal \(L^2\) and \(H(\operatorname{curl})\) approximations, so called fine-scale correctors must be added. The correctors show exponential decay and can therefore be truncated to local patches of macroscopic elements, so that it can be computed efficiently. This technique of numerical homogenization is known as Localized Orthogonal Decomposition (LOD) and well established for diffusion problems. The LOD framework decomposes the solution space into a coarse finite-dimensional space (spanned by standard finite element functions) and a fine-scale space, expressed as the kernel of a suitable interpolation/projection operator. A generalized finite element basis is constructed by adding corrections from the fine-scale space to the standard basis functions. An example of such a generalized basis function is displayed in Figure 2. These corrections are computed as solutions of a PDE on a fine grid. In this spirit, this part of the project can be seen as an extension of periodic homogenization results to more general rapidly varying coefficients or as an application of the LOD framework to a new problem class, namely \(H(\operatorname{curl})\)-elliptic problems.
Figure 2: Generalized basis function.
In ongoing work for the diffusion problem, we also devise multiscale techniques for the case of stochastic diffusion coefficients, that is we consider elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The method compresses the random partial differential operator to an effective quasilocal deterministic operator that represents the expected solution on a coarse scale of interest. Our analysis aims at quantifying the impact of uncertainty in the diffusion coefficient on the expected effective response of the process. As a further application, we also study the stabilizing effects of the discussed technique for time-harmonic elastodynamics. A simulation result is displayed in Figure 3.
We derived the FE-HMM for the above time-dependent and non-local model, analysed the semidiscrete error and showed the expected error bounds using a unified error analysis
We were able to show that, in a stationary \(H(\operatorname{curl})\) elliptic problem without scale separation, the exact solution can indeed be decomposed into a coarse and fine part, and established a full stability and convergence analysis of an LOD method for such problems [GHV18]
We proposed a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. We provided error estimates consisting of a priori and a posteriori terms and explored connections with existing homogenization theories [GP17], [GP19].
We established wavenumber explicit stability bounds for the elastic wave equation [BG16]
Future work
Analyse the full discrete error using a suitable time stepping method
Implementation of the derived scheme
LOD for the Maxwell case with randomly varying coefficients
Draw connections between LOD and the analytical (stochastic) homogenization theory
