# Math Technical Report Series

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Item Innovative Solution of a 2-D Elastic Transmission Problem(2006-07-12T14:01:23Z) Hsiao, George C.; Nigam, Nilima; Sändig, Anna-MargareteThis paper is concerned with a boundary-field equation approach to a class of boundary value problems exterior to a thin domain. A prototype of this kind of problems is the interaction problem with a thin elastic structure. We are interested in the asymptotic behavior of the solution when the thickness of the elastic structure approaches to zero. In particular, formal asymptotic expansions will be developed, and their rigorous justification will be considered. As will be seen, the construction of these formal expansions hinges on the solutions of a sequence of exterior Dirichlet problems, which can be treated by employing boundary element methods. On the other hand, the justification of the corresponding formal procedure requires an independence on the thickness of the thin domain for the constant in the Korn inequality. It is shown that in spite of the reduction of the dimensionality of the domain under consideration, this class of problems are in general not singular perturbation problems, because of appropriate interface conditions.Item A Newton-Imbedding Procedure for Solutions of Semilinear Boundary Value Problems in Sobolev Spaces(2006-07-12T13:54:38Z) Hsiao, G.C.This paper is concerned with the application of the Newton-imbedding iteration procedure to nonlinear boundary value problems in Sobolev spaces. A simple model problem for the second-order semilinear elliptic equations is considered to illustrate the main idea. The essence of the method hinges on the a priori estimates of solutions of the associated linear problem in appropriate Sobolev spaces. It is to our surprise that H1(Ω)-solution is not smooth enough to guarantee the convergence of the sequence generated by the procedure. Existence and uniqueness of solution to the original nonlinear problem are established constructively. An application of this approach to the Lamé system with nonlinear body force and its generalization to contain a nonlinear surface traction in elasticity will also be discussed.Item An Initial-Boundary Value Problem for the Viscous Compressible Flow(2006-07-07T14:03:08Z) Hebeker, Friedrich-Karl; Hsiao, George CA constructive approach is presented to treat an initial boundary value problem for isothermal Navier Stokes equations. It is based on a characteristics (Lagrangean) approximation locally in time and a boundary intergral equation method via nonstationary potentials. As a basic problem, the later leads to a Volterra integral equation of the first kind which is proved to be uniquely solvable and even coercive in some anistropic Sobolev spaces. The solution depends continuously upon the data and may be constructed by a quasioptimal Galerkin procedure.Item On the Boundary Integral Equation Method for a Mixed Boundary Value Problem of the Biharmonic Equation(Department of Mathematical Sciences, 2005) Cakoni, F.; Hsiao, G.C.; Wendland, W.Item Flow of a Model for Dilute Wormlike Micellar Solutions(Depatment of Mathematical Sciences, 2006) Rossi, L.F.; McKinley, G.; Cook, L.PItem Evaluation of the biot-savart integral for deformable elliptical gaussian vortex elements(Mathematical Sciences Department, 2005) Rossi, Louis F.This paper introduces two techniques for approximating the Biot-Savart integral for deforming elliptical Gaussian functions. The primary motivation is to develop a high spatial accuracy vortex method. The first technique is a regular perturbation of the streamfunction in the small parameter ε = (a-1)/(a+1)where a^2 is the aspect ratio of the basis function. This perturbative technique is suitable for direct interactions. In the far field, the paper studies the applicability of the fast multipole method for deforming elliptical Gaussians since the multipole series are divergent. The noncompact basis functions introduce a new computational length scale that limits the efficiency of the multipole algorithm but by imposing a lower bound on the finest mesh size, one can approximate the far-field stream function to any specified tolerance.Item On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients(Department of Mathematical Sciences, 2005-02-28) Mytnik, Leonid; Perkins, Edwin; Sturm, AnjaWe consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on R d for d greater than or equal to 1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Holder continuity of the solutions near zero is established.Item STABILITY OF VISCOELASTIC FILAMENTS: COMPARISON OF CONSTITUTIVE MODELS(Department of Mathematical Sciences, 1999-07-23) Olagunju, David O.A thin filament model is used to analyze the stability of a viscoelastic thread subject to uniaxial stretching. Linear stability analysis is carried out for a number of different constitutive models, namely the Johnson{Segalman,Giesekus, Phan{Thien Tanner, and FENE{CR. Our analysis shows that stability is controlled by the competing effects of surface tension which is destabilizing and axial normal stress which is stabilizing. Numerical simulations of the model equations are used to check the prediction of linear analysis. Results obtained agree with experimental observations.Item A mathematical and experimental study of ant foraging line dynamics(Department of Mathematical Sciences, 2005-04-13) Rossi, Louis F.; Johnson, KatieIn this article, we present a mathematical model coupled to an experimental study of ant foraging lines. Our laboratory experiments do not support the common traffic modeling assumption that ant densities and velocities are directly correlated. Rather, we find that higher order effects play a major role in observed behavior, and our model reflects this by including inertial terms in the evolution equation. A linearization of the resulting system yields left- and right-moving waves, in agreement with laboratory measurements. The linearized system depends upon two Froude numbers reflecting a ratio of the energy stored in the foraging line to the kinetic energy of the ants. Furthermore, the model predicts and the measurements support the existence of two distinct phase velocities.Item Eigenvalue stability of radial basis function discretizations for time-dependent problems(Department of Mathematical Sciences, 2005) Platte, R.B.; Driscoll, Tobin A.Differentiation matrices obtained with infinitely smooth radial basis function (RBF) collo- cation methods have, under many conditions, eigenvalues with positive real part, preventing the use of such methods for time-dependent problems. We explore this difficulty at theoretical and practical levels. Theoretically, we prove that differentiation matrices for conditionally positive definite RBFs are stable for periodic domains. We also show that for Gaussian RBFs, special node distributions can achieve stability in 1-D and tensor-product nonperiodic domains. As a more practical approach for bounded domains, we consider differentiation matrices based on least-squares RBF approximations and show that such schemes can lead to stable methods on less regular nodes. By separating centers and nodes, least-squares techniques open the possibility of the separation of accuracy and stability characteristics.Item Accurate Discretisation of a Nonlinear Micromagnetic Problem(Department of Mathematical Sciences, 2000) Monk, Peter B.; Vacus, O.In this paper we propose a finite element discretization of the Maxwell-Landau-Lifchitz-Gilbert equations governing the electromagnetic field in a ferromagnetic material. Our point of view is that it is desirable for the discrete problem to possess conservation properties similar to the continuous system. We first prove the existence of a new class of Liapunov functions for the continuous problem, and then for a variational formulation of the continuous problem. We also show a special continuous dependence result. Then we propose a family of mass-lumped finite element schemes for the problem. For the resulting semi-discrete problem we show that magnetization is conserved and that semi-discrete Liapunov functions exist. Finally we show the results of some computations that show the behavior of the fully discrete Liapunov functions.Item Recent Developments in Inverse Acoustic Scattering Theory(Department of Mathematical Sciences, 2000) Colton, David; Coyle, J.; Monk, Peter B.We survey some of the highlights of inverse scattering theory as it has developed over the past fifteen years, with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves. Included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects.Item Scattering of Time-Harmonic Electromagnetic Waves by Anisotropic Inhomongeneous Scatters or Impenetrable Obstacles(Department of Mathematical Sciences, 2000) Monk, Peter B.; Coyle, J.We investigate an overlapping solution technique to compute the scattering of time-harmonic electromagnetic waves in two dimensions. The technique can be used to compute waves scattered by penetrable anisotropic inhomogeneous scatterers or impenetrable obstacles. The major focus is on implementing the method using finite elements. We prove existence of a unique solution to the disctretized problem and derive an optimal convergence rate for the scheme, which is verified numerical by examples.Item Finite Element Method for Approximating Electro-Magnetic Scattering from a Conducting Object(Department of Mathematical Sciences, 2000) Kirsch, A.; Monk, Peter B.We provide an error analysis of a fully discrete finite element – Fourier series method for approximating Maxwell’s equations. The problem is to approximate the electromagnetic field scattered by a bounded, inhomogeneous and anisotropic body. The method is to truncate the domain of the calculation using a series solution of the field away from this domain. We first prove a decomposition for the Poincare-Steklov operator on this boundary into an isomorphism and a compact perturbation. This is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem. Using this decomposition, and edge elements to discretize the interior problem, we prove an optimal error estimate for the overall problem.Item Integral Representation in the Hodograph Plane for Compressible flow Problems(Department of Mathematical Sciences, 1999-12-07) Hanson, E.B.; Hsiao, George C.Compressible flow is considered in the hodograph plane. The fact that the equation for the stream function is linear there is exploited to derive a representation formula for the stream function, involving boundary data only, and a fundamental solution to the equation. For subsonic flow, an efficient algorithm for computation of the fundamental solution is also developed.Item Toward the Direct Analytical Determination of the Pareto Optima of a Differentiable Mapping, I: Domains in Finite-Dimensional Spaces(Department of Mathematical Sciences, 2000) Dallas, A.G.The problem of locating the Pareto-optimal points of a differentiable mapping $F: {\mathcal M}^N \to {I\kern-.30em R}^n$ is studied, with the domain ${\cal M}^N$ a differentiable N-dimensional submanifold-without-boundary in a euclidean space ${I\kern-.30em R}^{N_{0}}$ and $N_0 \ge N \ge n$. The case in which the domain is the closure of a bounded, regular, open subset of ${I\kern-.30em R}^N$ is also discussed. The search is initiated from these observations: for a manifold-domain, (1) the image of any Pareto optimum lies in the boundary of the range of F; (2) a point of the boundary of the range of F that also lies in the range must be the image of a singular point of F, i.e., must appear amongst the singular values of the map. Further conditions are then needed to distinguish which of the singular values should be discarded because they belong to the interior of the range; local tests of this sort are given for the bicriterial case (n = 2). A search procedure based on the present developments can systematically determine all of the Pareto optima for sufficiently simple F. The conditions established here may be regarded as analogues of the classical ones for the determination of the global extrema of a real-valued differentiable function. The results proven are illustrated with single examples, including plots of the ranges, singular points, and singular values.Item On the Convergence and Numerical Stability of the Second Waterman Scheme for Approximation of the Acoustic Field Scattered by a Hard Object(Department of Mathematical Sciences, 2000) Dallas, A.G.The numerical schemes of P.C. Waterman (J. Acoust. Soc. Am.45 (1969), 1417-1429), frequently referred to under the name of "the T-Matrix method," have formed the basis for many scattering computations in many settings. However, no successful analyses of the algorithms have been published, so the limitations on their range of applicability and numerical stability remain largely unknown; this is of particular importance because of the apparently inconsistent success achieved in numerical experiments. Here, we give an operator condition that guarantees the viability of the algorithm and mean-square convergence of the far-field patterns generated by the second Waterman scheme for the case of time-harmonic acoustic scattering by a hard obstacle; we prove further that the operator condition holds at least whenever the scattering obstacle is ellipsoidal. For the convergence proof, we also assume that the square of the wavenumber is not an interior Dirichlet eigenvalue for the negative Laplacianl in the contrary case, we show that the algorithm is at best numerically ill-coordinated. With this and previous experience in numerical applications, it appears that the performance of the algorithm is markedly shape-dependent; for certain obstacles, e.g., ellipsoids, instabilities are so localized in wavenumber that they are practically numerically irrelevant, while it is not clear whether the erratic results found in applications to various other shapes arise from a failure of convergence or form numerical instability.Item Basis Properties of Traces and Normal Derivatives of Spherical-Separable Solutions of the Helmholtz Equation(Department of Mathematical Sciences, 2000) Dallas, A.G.The classical solutions of the Heimholtz equation resulting from the separation-of-variables procedure in spherical coördinates are frequently used in one way or another to approximate other solutions. In particular, traces and/or normal derivatives of certain sequences of these spherical-separable solutions are commonly used as trial-and-test-functions in Galerkin procedures for the approximate solution of boundary-operator problems arising from the reformulation of exterior or interior boundary-value problems and set on the boundary Γ of the domain where a solution is wanted. While the completeness properties of these traces and normal derivatives in the usual Hilbert space L2( Γ) are well known, their basis properties are not. We show that such sequences of traces or normal derivatives of the outgoing spherical-separable solutions form bases for L2( Γ) only when Γ is a sphere centered at the pole of the spherical solutions; corresponding results are given for the entire solutions, accounting for the possibility of an interior eigenvalue. We identify other Hilbert spaces, connected with the far-field pattern, for which these functions do provide bases. We apply the results to discuss some aspects of the Waterman schemes for approximate solutions of scattering problems (the so-called “T-matrix method”), including the previous article of KRISTENSSON, RAMM, and STRÖM (J. Math. Phys.24 (1983), 2619-2631) on the convergence of such methods.Item Hybrid Coupled Finite-Boundary Element Methods for Elliptical Systems of Second Order(Department of Mathematical Sciences, 2000) Hsiao, George C.; Schnack, E.; Wendland, W.L.In this hybrid method, we consider, in addition to traditional finite elements, the Trefftz elements for which the governing equations of equilibrium are required to be satisfied a priori within the subdomain elements. If the Trefftz elements are modelled with boundary potentials supported by the individual element boundaries, this defines the so–called macro–elements. These allow one to handle in particular situations involv-ing singular features such as cracks, inclusions, corners and notches providing a locally high resolution of the desired stress fields, in combination with a traditional global varia-tional FEM analysis. The global stiffness matrix is here sparse as the one in conventional FEM. In addition, with slight modifications, the macro–elements can be incorporated into standard commercial FEM codes. The coupling between the elements is modelled by using a generalized compatibility condition in a weak sense with additional elements on the skeleton. The latter allows us to relax the continuity requirements for the global displacement field. In particular, the mesh points of the macro–elements can be chosen independently of the nodes of the FEM structure. This approach permits the combination of independent meshes and also the exploitation of modern parallel computing facilities. We present here the formulation of the method and its functional analytic setting as well as corresponding discretizations and asymptotic error estimates. For illustration, we include some computational results in two– and three–dimensional elasticity.Item Domain Decomposition Methods via Boundary Integral Equations(Department of Mathematical Sciences, 2000-12-21) Hsiao, George C.; Steinbach, O.; Wendland, W.L.Domain decomposition methods are designed to deal with coupled or transmission problems for partial differential equations. Since the original boundary value problem is replaced by local problems in substructures, domain decomposition methods are well suited for both parallelization and coupling of different discretization schemes. In general, the coupled problem is reduced to the Schur complement equation on the skeleton of the domain decomposition. Boundary integral equations are used to describe the local Steklov-Poincare operators which are basic for the local Dirichlet-Neumann maps. Using different representations of the Steklov-Poincare operators we formulate and analyze various boundary element methods employed in local discretization schemes. We give sufficient conditions for the global stability and derive corresponding a priori error estimates. For the solution of the resulting linear systems we describe appropriate iterative solution strategies using both local and global preconditioning techniques.