A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed. Optimal order error estimates are established in various norms, and nodal superconvergence is demonstrated. Discretization in time by explicit single-step methods is discussed.

This paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps. A weak formulation of the problem is first presented, which then yields the existence and uniqueness of a solution to the problem by classical methods. It is shown that the application of the ghost fluid method by Fedkiw, Kang, and Liu to this problem can ...

Most common implementations of the NEGF approach are done using the Finite-Difference Method (FDM) on uniform grids. This discretization scheme allows one to study many types of planar devices, such as MOSFETs, superlattices and planar heterojunctions. Within this paradigm, carrier scattering can be included through the selfenergy approach, almost always under the assumption of local interactio...

We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each ...

In this paper, we first describe a fourth order accurate finite difference discretization for both the Laplace equation and the heat equation with Dirichlet boundary conditions on irregular domains. In the case of the heat equation we use an implicit time discretization to avoid the stringent time step restrictions associated with explicit schemes. We then turn our focus to the Stefan problem a...

If reciprocity principles are invoked in seismic data processing algorithms, one has to make sure that those processes maintain reciprocity. I outline briefly the derivation of reciprocity principles for elastic wave equations and show that discretized or approximated wave equations can lose symmetry properties and thus reciprocities. This effect is important if one is concernd about true-ampli...

In this paper we consider a class of unfitted finite element methods for discretization of partial differential equations on surfaces. In this class of methods known as the Trace Finite Element Method (TraceFEM), restrictions or traces of background surface-independent finite element functions are used to approximate the solution of a PDE on a surface. We treat equations on steady and time-depe...

We present a fast penetration depth estimation algorithm between deformable polyhedral objects. We assume the continuum of non-rigid models are discretized using standard techniques, such as finite element or finite difference methods. As the objects deform, the distance fields are deformed accordingly to estimate penetration depth, allowing enforcement of non-penetration constraints between tw...

How to properly specify boundary conditions for pressure is a longstanding problem for the incompressible Navier–Stokes equations with no-slip boundary conditions. An analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the Laplacian and Leray projection operators. Here we make use of this formula to (a) improve the accuracy ...