This paper is concerned with an interior penalty discontinuous Galerkin (IPDG) method based on a flexible type of non-polynomial local approximation space for the Helmholtz equation with varying wavenumber. The local approximation space consists of multiple polynomial-modulated phase functions which can be chosen according to the phase information of the solution. We obtain some approximation p...

The purpose of this paper is to approximate the solution of a Cauchy integral equation of the second kind, using projection, finite rank approximations and a regularization procedure. We us the Kantorovich projection, the Sloan projection, the Galerkin projection, respectively. We give a general framework and we prove the existence of the solution for a projection schemes. 2012 Elsevier Inc. Al...

We continue the development of high-order accurate thin layer approximations for time-domain electromagnetics and focus in this paper on thin transmission layers. The thin transmission layer approximations are valid for general isotropic materials and certain classes of anisotropic materials. The models also allow the inclusion of smoothly curved layers. Both dielectric and magnetic materials c...

We study a semi-discrete Galerkin method for solving the single-layer equation Vu = f with an approximating subspace of piecewise constant functions. Error bounds in Sobolev norms kkk s with ?1 s < 1 2 are proven and are of the same order as for the original Galerkin method. The distinctive features of the present work are that we handle irregular meshes and do not rely on Fourier methods. The ...

In this work, we consider the forced generalized Burgers-Huxley equation and establish existence uniqueness of a global weak solution using Faedo-Galerkin approximation method. Under smoothness assumptions on initial data external forcing, also obtain further regularity results solutions. Taking forcing to be zero, positivity result as well bound classical are estab...

We consider linear first order scalar equations of the form ρt + div(ρv) + aρ = f with appropriate initial and boundary conditions. It is shown that approximate solutions computed using the discontinuous Galerkin method will converge in L[0, T ;L(Ω)] when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions. In particula...

We consider a stochastic Galerkin and collocation discretization scheme for solving elliptic PDEs with random coefficients and forcing term, which are assumed to depend on a finite, but possibly large number of random variables. Both methods consist of a hierarchic wavelet discretization in space and a sequence of hierarchic approximations to the law of the random solution in probability space....

Complementarity problems and variational inequalities arise in a wide variety of areas, including machine learning, planning, game theory, and physical simulation. In all of these areas, to handle large-scale problem instances, we need fast approximate solution methods. One promising idea is Galerkin approximation, in which we search for the best answer within the span of a given set of basis f...

Optimal control and optimal shape design problems for the Navier-Stokes equations arise in many important practical applications, such as design of optimal profiles [7], drag minimization [9], [11], and heating and cooling [12], among others. Typically, optimal control problems for the NavierStokes equations combine Lagrange multiplier techniques to enforce the constraints and to derive an opti...