Discontinuous galerkin finite elements method with structure preserving time integrators for gradient flow equations

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2015
Sarıaydın Filibelioğlu, Ayşe
Gradient flows are energy driven evolutionary equations such that the energy decreases along solutions. There have been surprisingly a large number of well-known partial differential equations (PDEs) which have the structure of a gradient flow in different research areas such as fluid dynamics, image processing, biology and material sciences. In this study, we focus on two systems which can be modeled by gradient flows;Allen-Cahn and Cahn-Hilliard equations. These equations model the phase separation in material science. Since an essential feature of the Allen-Cahn and Cahn-Hilliard equations is the energy decreasing property, it is important to design efficient and accurate numerical schemes that satisfy the corresponding energy decreasing property. We have used symmetric interior penalty Galerkin (SIPG) method to discretize the Allen-Cahn and Cahn-Hilliard equations in space. The resulting large system of ordinary differential equations (ODEs) as a gradient system are solved by the energy stable (energy decreasing) time integrators: implicit Euler and average vector field (AVF) methods. We have shown that implicit Euler and AVF time integrators coupled with SIPG method are unconditionally energy stable. Numerical results for both equations with polynomial and logarithmic energy functions, and constant and variable mobility functions illustrate the efficiency and accuracy of this approach. Advective Allen-Cahn equation is the simplest model of surface tension in the droplet breakup phenomena. The small surface time scale and convective time scale lead to unphysical oscillations in the solution. In contrast to the discretization of Allen-Cahn and Cahn-Hilliard equations using the method of lines, the advective Allen-Cahn equation is first discretized in time using implicit Euler method and the resulting sequence of semi–linear elliptic equations are solved with an adaptive algorithm. This corresponds to Rothe’s method. As a remedy of unphysical oscillations, an adaptive version of SIPG method based on residual based a posteriori error estimate is applied. Numerical results for convection dominated Allen-Cahn equation show the performance of adaptive algorithm.

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Citation Formats
A. Sarıaydın Filibelioğlu, “Discontinuous galerkin finite elements method with structure preserving time integrators for gradient flow equations,” Ph.D. - Doctoral Program, Middle East Technical University, 2015.