Christopher B. Davis

A Partition of Unity Method with Penalty for Fourth Order Problems

A partition of unity method for fourth order problems is proposed. As a model problem, we focus on the biharmonic problem with either clamped or simply supported boundary conditions when the domain is a bounded polygon. The algorithm is presented, error estimates are made, and numerical results are shown to verify the error estimates.

Meshfree Particle Methods in the Framework of Boundary Element Methods for the Helmholtz Equation

In this paper, we study electromagnetic wave scattering from periodic structures and eigenvalue analysis of the Helmholtz equation. Boundary element method (BEM) is an effective tool to deal with Helmholtz problems on bounded as well as unbounded domains. Recently, Oh et al. (Comput. Mech. 48:27–45, 2011) developed reproducing polynomial boundary particle methods (RPBPM) that can handle effectively boundary integral equations in the framework of the collocation BEM. The reproducing polynomial particle (RPP) shape functions used in RPBPM have compact support and are not periodic. Thus it is not ideal to use these RPP shape functions as approximation functions along the boundary of a circular domain. In order to get periodic approximation functions, we consider the limit of the RPP shape function as its support is getting infinitely large. We show that the basic approximation function obtained by the limit of the RPP shape function yields accurate solutions of Helmholtz problems on circular, or annular domains as well as on the infinite domains.

A Partition of Unity Method for the Obstacle Problem of Simply Supported Kirchhoff Plates

We consider a partition of unity method (PUM) for the displacement obstacle problem of simply supported Kirchhoff plates. We show that this method converges optimally in the energy norm on general polygonal domains provided that appropriate singular enrichment functions are included in the approximation space. The performance of the method is illustrated by numerical examples.

A partition of unity method for the displacement obstacle problem of clamped Kirchhoff plates

A partition of unity method for the displacement obstacle problem of clamped Kirchhoff plates is considered in this paper. We derive optimal error estimates and present numerical results that illustrate the performance of the method.

A mixed formulation of the Stefan problem with surface tension

A dual formulation and finite element method is proposed and analyzed for simulating the Stefan problem with surface tension. The method uses a mixed form of the heat equation in the solid and liquid (bulk) domains, and imposes a weak formulation of the interface motion law (on the solidliquid interface) as a constraint. The basic unknowns are the heat fluxes and temperatures in the bulk, and the velocity and temperature on the interface. The formulation, as well as its discretization, is viewed as a saddle point system. Well-posedness of the time semi-discrete and fully discrete formulations is proved in three dimensions, as well as an a priori (stability) bound and conservation law. Simulations of interface growth (in two dimensions) are presented to illustrate the method.