Matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the num...

Complementary problems play a central role in equilibrium finding, physical simulation, and optimization. As a consequence, we are interested in understanding how to solve these problems quickly, and this often involves approximation. In this paper we present a method for approximately solving strictly monotone linear complementarity problems with a Galerkin approximation. We also give bounds f...

An error analysis is presented for the spectral-Galerkin method to the Helmholtz equation in 2and 3-dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator, and then a spectral-Galerkin method is employed to approximate the reduced problem. The error analysis is based on exploring delicate asymptotic ...

In this paper, the dynamic response of a piezoelectric transducer in contact with an acoustic media is formulated. This serves as a model problem for an ultrasonic phased array in contact with the human body. These arrays may be used for hyperthermia or ablation surgery; the focus of this paper, however, is the eecient numerical formulation for modeling the array problem. A nite element formula...

In this article we study the properties of the hyperinterpolation operator on the unit disk D in R, approximating the orthogonal projection of a function onto the family of polynomials of degree n. A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results then prove the uniform convergence of the hyperinterpolation approximation of functions in the class ...

In this paper, we develop the locally divergence-free discontinuous Galerkin method for numerically solving the Maxwell equations. The distinctive feature of the method is the use of approximate solutions that are exactly divergence-free inside each element. As a consequence, this method has a smaller computational cost than that of the discontinuous Galerkin method with standard piecewise poly...

A homotopic Galerkin approach to the solution of the Fokker-Planck-Kolmogorov equation is presented. It is argued that the ideal Hilbert Space to approximate the exact solution, ψ∗, is the space L2(dψ) where dψ∗ is the probability measure induced on <n by the solution ψ∗ itself. Since this space is not accessible, it is shown that given an initial approximation sufficiently close to the exact s...

We propose and study an adaptive version of the discontinuous Galerkin method for Hamilton–Jacobi equations. It works as follows. Given the tolerance and the degree of the polynomial of the approximate solution, the adaptive algorithm finds a mesh on which the approximate solution has an L1-distance to the viscosity solution no bigger than the prescribed tolerance. The algorithm uses three main...