DECAY OF POSITIVE WAVES OF HYPERBOLIC BALANCE LAWS

Historically, decay rates have been used to provide quantitative and quali- tative information on the solutions to hyperbolic conservation laws. Quantitative results include the establishment of convergence rates for approximating procedures and numer- ical schemes. Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time. This work presents two decay estimates on the positive waves for systems of hyperbolic and gen- uinely nonlinear balance laws satisfying a dissipative mechanism. The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics [7, 17, 241.

A Practical Algorithm for Determining the Optimal Pseudo-Boundary in the Method of Fundamental Solutions

One of the main difficulties in the application of the method of fundamental solutions(MFS)is the determination of the position of the pseudo-boundary on which are placed the singularities in terms of which the approximation is expressed.In this work,we propose a simple practical algorithm for determining an estimate of the pseudo-boundary which yields the most accurate MFS approximation when the method is applied to certain boundary value problems.Several numerical examples are provided.

The Pressure-Streamfunction MFS Formulation for the Detection of an Obstacle Immersed in a Two-Dimensional Stokes Flow

In this paper we consider a geometric inverse problem which requires detecting an unknown obstacle such as a submarine or an aquatic mine immersed in a Stokes slow viscous stationary flow of an incompressible fluid,from a single set of Cauchy(fluid velocity and stress force)boundary measurements.The numerical reconstruction is based on the method of fundamental solutions(MFS)for the pressure and streamfunction in two dimensions combined with regularization.Numerical results are presented and discussed.

Spectral Element Discretization of the Stokes Equations in Deformed Axisymmetric Geometries

In this paper,we study the numerical solution of the Stokes system in deformed axisymmetric geometries.In the azimuthal direction the discretization is carried out by using truncated Fourier series,thus reducing the dimension of the problem.The resulting two-dimensional problems are discretized using the spectral element method which is based on the variational formulation in primitive variables.The meridian domain is subdivided into elements,in each of which the solution is approximated by truncated polynomial series.The results of numerical experiments for several geometries are presented.

Efficient Algorithms for Approximating Particular Solutions of Elliptic Equations Using Chebyshev Polynomials

In this paper,we propose efficient algorithms for approximating particular solutions of second and fourth order elliptic equations.The approximation of the particular solution by a truncated series of Chebyshev polynomials and the satisfaction of the differential equation lead to upper triangular block systems,each block being an upper triangular system.These systems can be solved efficiently by standard techniques.Several numerical examples are presented for each case.

Galerkin Formulations of the Method of Fundamental Solutions

In this paper,we introduce two Galerkin formulations of theMethod of Fundamental Solutions(MFS).In contrast to the collocation formulation of the MFS,the proposed Galerkin formulations involve the evaluation of integrals over the boundary of the domain under consideration.On the other hand,these formulations lead to some desirable properties of the stiffness matrix such as symmetry in certain cases.Several numerical examples are considered by these methods and their various features compared.

The Method of Fundamental Solutions for Steady-State Heat Conduction in Nonlinear Materials

The steady-state heat conduction in heat conductors with temperature dependent thermal conductivity and mixed boundary conditions involving radiation is investigated using the method of fundamental solutions.Various computational issues related to the method are addressed and numerical results are presented and discussed for problems in two and three dimensions.

Analytic Regularity for a Singularly Perturbed Reaction-Convection-Diffusion Boundary Value Problem with Two Small Parameters

We consider a second order,two-point,singularly perturbed boundary value problem,of reaction-convection-diffusion type with two small parameters,and we obtain analytic regularity results for its solution,under the assumption of analytic input data.First,we establish classical differentiability bounds that are explicit in the order of differentiation and the singular perturbation parameters.Next,for small values of these parameters we show that the solution can be decomposed into a smooth part,boundary layers at the two endpoints,and a negligible remainder.Derivative estimates are obtained for each component of the solution,which again are explicit in the differentiation order and the singular perturbation parameters.

On C-Bases, Partition Pairs and Filtrations for Induced or Restricted Specht Modules

We obtain alternative explicit Specht filtrations for the induced and the restricted Specht modules in the Hecke algebra of the symmetric group (defined over the ring A = Z[g^1/2, q^-1/2], where q is an indeterminate) using C-bases for these modules. Moreover, we provide a link bet ween a certain C-basis for the induced Specht module and the notion of pairs of partitions.

AN hp-FEM FOR SINGULARLY PERTURBED TRANSMISSION PROBLEMS

We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the hp-Finite Element Method （FEM） on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also nresented.

Local RBF Algorithms for Elliptic Boundary Value Problems in Annular Domains

A local radial basis function method(LRBF)is applied for the solution of boundary value problems in annular domains governed by the Poisson equation,the inhomogeneous biharmonic equation and the inhomogeneous Cauchy-Navier equations of elasticity.By appropriately choosing the collocation points we obtain linear systems in which the coefficient matrices possess block sparse circulant structures and which can be solved efficiently using matrix decomposition algorithms(MDAs)and fast Fourier transforms(FFTs).The MDAs used are appropriately modified to take into account the sparsity of the arrays involved in the discretization.The leave-one-out cross validation(LOOCV)algorithm is employed to obtain a suitable value for the shape parameter in the radial basis functions(RBFs)used.The selection of the nearest centres for each local influence domain is carried out using a modification of the kdtree algorithm.In several numerical experiments,it is demonstrated that the proposed algorithm is both accurate and capable of solving large scale problems.

Preface

A special session on Numerical Analysis,in honor of Professor Graeme Fairweather’s 70th birthday This special issue of Advances in Applied Mathematics and Mechanics consists of a se-lection of invited and contributed papers that were presented in a Special Session of the 8th International Conference on Scientific Computing and Applications that was held at the University of Nevada,Las Vegas,from April 1 to 4,2012.The Special Session was organized in order to celebrate the 70th birthday of Professor Graeme Fairweather.

A Moving Pseudo-Boundary MFS for Three-Dimensional Void Detection

We propose a new moving pseudo-boundary method of fundamental solutions(MFS)for the determination of the boundary of a three-dimensional void(rigid inclusion or cavity)within a conducting homogeneous host medium from overdetermined Cauchy data on the accessible exterior boundary.The algorithm for imaging the interior of the medium also makes use of radial spherical parametrization of the unknown star-shaped void and its centre in three dimensions.We also include the contraction and dilation factors in selecting the fictitious surfaces where the MFS sources are to be positioned in the set of unknowns in the resulting regularized nonlinear leastsquares minimization.The feasibility of this new method is illustrated in several numerical examples.