Numerical Analysis of Evolution Problems in Multiphysics

Abstract: In this thesis we study numerical methods for evolution problems in multiphysics. The term multiphysics is commonly used to describe physical phenomena that involve several interacting models. Typically, such problems result in coupled systems of partial differential equations. This thesis is essentially divided into two parts, which address two different topics with applications in multiphysics. The first topic is numerical analysis for multiscale problems, with a particular focus on heterogeneous materials, like composites. For classical finite element methods such problems are known to be numerically challenging, due to the rapid variations in the data. One of our main goals is to develop a numerical method for the thermoelastic system with multiscale coefficients. The method we propose is based on the localized orthogonal decomposition (LOD) technique introduced in Målqvist and Peterseim (Math Comput 83(290):2583-2603, 2014). This is performed in three steps, first we extend the LOD framework to parabolic problems (Paper I) and then to linear elasticity equations (Paper II). Using the theory developed in these two papers we address the thermoelastic system (Paper III). In addition, we aim to extend the LOD framework to differential Riccati equations where the state equation is governed by a multiscale operator. The numerical solution of such problems involves solving many parabolic equations with multiscale coefficients. Hence, by applying the method developed in Paper I to Riccati equations the computational gain may be significantly large. In this thesis we show that this is indeed the case (Paper IV). The second part of this thesis is devoted to the Joule heating problem, a coupled nonlinear system describing the temperature and electric current in a material. Analyzing this system turns out to be difficult due to the low regularity of the nonlinear term. We overcome this issue by introducing a new variational formulation based on a cut-off functional. Using this formulation, we prove (Paper V) strong convergence of a large class of finite element methods for the Joule heating problem with mixed boundary conditions on nonsmooth domains in three dimensions.