Local discontinuous Galerkin method for phase transition problems

Abstract

In this thesis we develop a local discontinuous Galerkin (LDG) finite element method to solve mathematical models for phase transitions in solids and fluids. The first model we study is called a viscosity-capillarity (VC) system associated with phase transitions in elastic bars and Van der Waals fluids. We develop and analyze an LDG discretization for the VC system. We prove L2-stability for the VC system with a general stress- strain relation. Using a priori error analysis, we provide an error estimate for the LDG discretization of the VC system when the stress-strain relation is linear, assuming that the solution is sufficiently smooth and the system is in a hyperbolic region. Numerical examples are provided to verify the LDG discretization of the VC system. Secondly, we consider the (non)-isothermal Navier-Stokes-Korteweg (NSK) equations describing the dynamics of a compressible fluid with liquid-vapor phase change. The NSK equations contain a stress tensor related to the capillarity forces. The Van der Waals equation of state is used in the NSK equations to describe the pressure in both liquid and vapor states. We solve the NSK equations with the LDG method in conservative form without additional stabilization. In solving the ordinary differential equations resulting from the LDG discretization, we use a time-implicit Runge-Kutta method to circumvent a severe time restriction due to the third order derivative of the density. Computations demonstrate the accuracy, convergence and stability of the proposed LDG discretizations. Finally, a mesh adaptation algorithm is studied for the LDG discretization of the (non)-isothermal NSK equations to save computational cost and to capture the interface more accurately. The motivation is the fact that a fine resolution is only required in the interfacial region and that the LDG method is suited for hanging nodes. As an adaptation criterion, the locally largest value of the density gradient is used to select candidate elements for refinement and coarsening. Boundary conditions where vapor bubbles and a liquid droplet are in contact with a solid wall are considered in the numerical examples.