Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions

Abstract

We present a novel and efficient method for solving the Poisson equation, the heat equation, and Stefan-type problems with Robin boundary conditions over potentially moving, arbitrarily-shaped domains. The method utilizes a level set framework, thus it has all of the benefits of a sharp, implicitly-represented interface such as the ease of handling complex topological changes. This method is straightforward to implement and leads to a linear system that is symmetric and positive definite, which can be inverted efficiently with standard iterative methods. This approach is second-order accurate for both the Poisson and heat equations, and first-order accurate for the Stefan problem. We demonstrate the accuracy in the L 1 and L ∞ norms.