Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

Many problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus, we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully-discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.

Implicit Method and Stability of Dynamic Grid for Nonlinear Schroedinger equation

This work consists of two parts. The first part studying implicit method for Nonlinear Scrhoedinger equation, while the second part study the Backward Euler in dynamic Grid methods for the Nonlinear Schroedinger with critical power nonlinearity.A first-order implicit backward Euler method is studied for applicaton to nonlinear Schroedinger with critical powerdepicting blow-up. The focal point of the paper is to examine the efficiency and reliability of the method in handling of possibility of blow-up to capture the asymptotic behavior. Using field of values, we observe the distortion of eigen-values inclusion of spatial grid discretization.