Laplace's Equation and Network Flows

The Mixed Problem for Laplace's Equation in a Class of Lipschitz Domains

INTRODUCTION In this paper, we consider the mixed problem for Laplace's equation in a domain in. We assume that N and D give a decomposition of @. By this

Design method for electromagnetic cloak with arbitrary shapes based on Laplace's equation.

In transformation optics, the space transformation is viewed as the deformation of a material. The permittivity and permeability tensors in the transformed space are found to correlate with the deformation field of the material. By solving the Laplace's equation, which describes how the material will deform during a transformation, we can design electromagnetic cloaks with arbitrary shapes if t...

On the Numerical Solution of the Dirichlet Problem for Laplace's Difference Equation*

Integral Equation Methods for Solving Laplace's Equation with Nonlinear Boundary Conditions: the Smooth Boundary Case

A nonlinear boundary value problem for Laplace's equation is solved numerically by using a reformulation as a nonlinear boundary integral equation. Two numerical methods are proposed and analyzed for discretizing the integral equation, both using product integration to approximate the singular integrals in the equation. The first method uses the product Simpson's rule, and the second is based o...

Operator adapted spectral element methods I: harmonic and generalized harmonic polynomials

The paper presents results on the approximation of functions which solve an elliptic diierential equation by operator adapted systems of functions. Compared with standard polynomials, these operator adapted systems have superior local approximation properties. First, the case of Laplace's equation and harmonic polynomials as operator adapted functions is analyzed and sharp estimates for the app...