Local discontinuous Galerkin methods for parabolic interface problems with homogeneous and non-homogeneous jump conditions

Abstract

In this paper, we present the local discontinuous Galerkin (LDG) methods for solving second order parabolic interface problems in two-dimensional convex polygonal domains with homogeneous and non-homogeneous jump conditions, respectively. We start analyzing from the homogeneous problems. After giving the semi discrete LDG method, we derive its stability and a prior error estimate. Then, after a series of derivation, we obtain the LDG method for the non-homogeneous problems, which takes the same formulation as the homogeneous case with a special choice of numerical fluxes that incorporate the jump conditions across interface. So its analysis can be easily obtained within the framework developed for the homogeneous case. It is an advantage of the LDG method that it provides a natural framework to enforce the discontinuities in both the potential and the flux weakly in the discrete formulation provided that the triangulation of the domain is fitted to the interface. Numerical experiments are conducted to confirm our theoretical analysis with the help of the explicit temporal discretization and show that the LDG methods have the optimal and suboptimal convergence rate in the L2 norms for the potential and the flux, respectively. In addition, they also indicate that the LDG methods possess the same convergence properties in the L∞ norms. What is more, to avoid the strict time-step restriction of explicit schemes, the implicit integration factor (IIF) method is employed, which not only relaxes the time step to Δt=O(hmin) but also remains the important property of LDG method that the computation can proceed element by element and avoid solving a global system of algebraic equations as the standard implicit schemes do. We provide numerical examples to illustrate that the numerical scheme combined LDG method with IIF method indeed reduces the computational cost greatly.