Numerical methods for deterministic and stochastic differential equations with multiple scales and high contrasts

Mathematical models involving multiple scales are essential for the description of physical systems. In particular, these models are important for the simulation of time-dependent phenomena, such as the heat flow, where the Laplacian contains mixed and indistinguishable fast and slow modes. Stationary problems can also exhibit a multiscale nature. For example, elliptic equations governed by a diffusion coefficient with strong discontinuities have solutions characterized by regions with a high gradient. Simulating such models is very demanding, as the computational cost of standard numerical methods is usually ruled by the fastest dynamics or the smallest scale.

In the first part of this thesis, we develop multirate integration methods for deterministic and stochastic time-dependent problems with disparate time-scales. The cost of traditional schemes for such problems is prohibitive due to step size restrictions in the explicit case or solutions to large nonlinear systems in the implicit case. Existing multirate methods are either implicit or make use of interpolations, which trigger instabilities, or are based on a scale separation assumption, which is not satisfied by parabolic problems. Here we introduce a new framework based on modified equations which allows for the development of a whole new class of interpolation-free explicit multirate numerical methods, which do not need any scale separation, are stable and accurate. For deterministic problems, our methodology is based on the replacement of the original right-hand side by an averaged force, whose stiffness is reduced due to a fast but cheap auxiliary problem. Integrating the modified equation and the auxiliary problems by explicit schemes is generally cheaper than integrating the original problem. We thus introduce a multirate method based on stabilized explicit schemes and prove its efficiency, stability and accuracy. Numerical experiments show that standard schemes and our multirate approach provide essentially the same solutions; hence, the bottleneck caused by the stiffness of a few degrees of freedom is overcome without sacrificing accuracy. We also generalize the same framework to stochastic differential equations, where we need to introduce a damped diffusion term for which the resulting modified equation inherits the mean-square stability properties of the original problem. An interpolation-free stabilized explicit multirate method for stochastic equations is then derived.

In the second part of this thesis, we consider elliptic problems with high gradients and develop a local adaptive discontinuous Galerkin scheme. Local methods for such problems already exist in literature; however, they are usually based on iterations and have several downsides. In particular, their a priori error analysis is based on rather strong and nonphysical assumptions and they lack a rigorous a posteriori error analysis. The scheme that we propose is based on a coarse solution on the full domain which is subsequently improved by solving local elliptic problems only once on subdomains with artificial boundary conditions. The a priori error analysis is performed under minimal regularity assumptions due to the gradient discretization framework. Furthermore, we derive a posteriori error estimators based on conforming fluxes and potential reconstructions which can be used to identify the local subdomains on the fly, are free of undetermined constants and robust in singularly perturbed regimes.