# Numerical Analysis of Evolution Problems in Multiphysics Doktorsavhandling, 2018

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.

regularity

generalized finite element

localized orthogonal decomposition

multiscale

finite element method

Thermoelasticity

Riccati equations

thermistor

linear elasticity

Joule heating

parabolic equations

Pascal
Opponent: Olof Runborg, Department of Mathematics, Royal Institute of Technology (KTH), Stockholm

## Författare

Matematik

#### Multiscale techniques for parabolic equations

Numerische Mathematik,; Vol. 138(2018)p. 191-217

Artikel i vetenskaplig tidskrift

#### A multiscale method for linear elasticity reducing Poisson locking

Computer Methods in Applied Mechanics and Engineering,; Vol. 310(2016)p. 156-171

Artikel i vetenskaplig tidskrift

#### A Generalized Finite Element Method for Linear Thermoelasticity

Mathematical Modelling and Numerical Analysis,; Vol. 51(2017)p. 1145-1171

Artikel i vetenskaplig tidskrift

#### Max Jensen, Axel Målqvist, and Anna Persson - Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions

In many engineering applications it is important to perform computer simulations of various physical phenomena. For example, it is crucial to test how the design of an aircraft part effects the overall stability of the aircraft, before we put passengers in it.

To perform simulations we need numerical methods to approximate solutions to equations. However, to propose effective methods that generate approximations with small errors from the exact solution is often a difficult task.

This thesis focuses on numerical methods for evolution problems in multiphysics. In particular, we are interested in finding effective methods for materials that varies on a very fine scale, like fiber reinforced structures. Today's increasing interest and usage of such materials in, for instance, the aircraft industry poses a high demand of new effective numerical methods.

#### Ämneskategorier

Beräkningsmatematik

Matematisk analys

#### ISBN

978-91-7597-713-3

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4394

#### Utgivare

Chalmers tekniska högskola

Pascal

Opponent: Olof Runborg, Department of Mathematics, Royal Institute of Technology (KTH), Stockholm

2018-04-06