Heterogeneous Optimized Schwarz Methods for Coupling Helmholtz and Laplace Equations

Optimized Schwarz methods have been developed at the continuous level; in order to obtain optimized transmission conditions, the underlying partial differential equation (PDE) needs to be known. Classical Schwarz methods on the other hand can be used in purely algebraic form, which have made them popular. Their performance can however be inferior compared to that of optimized Schwarz methods. We present in this paper a discovery algorithm, which, based purely on algebraic information, allows us to obtain an optimized Schwarz preconditioner for a large class of numerically discretized elliptic PDEs. The algorithm detects the nature of the elliptic PDE, and then modifies a classical algebraic Schwarz preconditioner at the algebraic level, using existing optimization results from the literature on optimized Schwarz methods. Numerical experiments using elliptic problems discretized by Q 1-FEM, P 1-FEM, and FDM demonstrate the algebraic nature and the effectiveness of the discovery algorithm.

We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.

In this work we consider the optimized Schwarz method designed for computational domains that feature spherical or almost spherical interfaces. In the first part, we consider the diffusion-reaction problem. We provide a convergence analysis of the generalized Schwarz method and, following [G. Gigante and C. Vergara, Numer. Math., 131 (2015), pp. 369--404], we discuss an optimization procedure for constant interface parameters leading to a Robin--Robin scheme. Finally, we present some numerical results both in spherical and in ellipsoidal domains. In the second part of the work, we address the fluid-structure interaction problem. Again, we provide a convergence analysis and discuss optimized choices of constant interface parameters. Finally, we present three-dimensional numerical results inspired by hemodynamic applications, to validate the proposed choices in the presence of large added mass effect. In particular, we consider numerical experiments both in an ideal spherical domain and in a realistic abdominal aortic aneurysm.

Solving partial differential equations (PDEs) on infinite domains has been a challenging task in physical simulations and geometry processing. We introduce a general technique to transform a PDE problem on an unbounded domain to a PDE problem on a bounded domain. Our method uses the Kelvin Transform, which essentially inverts the distance from the origin. However, naive application of this coordinate mapping can still result in a singularity at the origin in the transformed domain. We show that by factoring the desired solution into the product of an analytically known (asymptotic) component and another function to solve for, the problem can be made continuous and compact, with solutions significantly more efficient and well-conditioned than traditional finite element and Monte Carlo numerical PDE methods on stretched coordinates. Specifically, we show that every Poisson or Laplace equation on an infinite domain is transformed to another Poisson (Laplace) equation on a compact region. In other words, any existing Poisson solver on a bounded domain is readily an infinite domain Poisson solver after being wrapped by our transformation. We demonstrate the integration of our method with finite difference and Monte Carlo PDE solvers, with applications in the fluid pressure solve and simulating electromagnetism, including visualizations of the solar magnetic field. Our transformation technique also applies to the Helmholtz equation whose solutions oscillate out to infinity. After the transformation, the Helmholtz equation becomes a tractable equation on a bounded domain without infinite oscillation. To our knowledge, this is the first time that the Helmholtz equation on an infinite domain is solved on a bounded grid without requiring an artificial absorbing boundary condition.

In this paper, we are interested in scalable Domain Decomposition Methods (DDM). To this end, we introduce and study a new Coarse Space Correction algorithm for Optimized Schwarz Methods(OSM) : the DCS-DGLC algorithm. The main idea is to use a Discontinuous Galerkin like formulation to compute a discontinuous coarse space correction. While the local spaces remain continuous, the coarse space should be discontinuous to compensate the discontinuities introduced by the OSM at the interface between neighboring subdomains. The discontinuous coarse correction algorithm can not only be used with OSM but also be used with any one-level DDM that produce discontinuous iterates. While ideas from Discontinuous Galerkin(DG) are used in the computation of the coarse correction, the final aim of the DCS-DGLC algorithm is to compute in parallel the discrete solution to the classical non-DG finite element problem.

A dual integral formulation for the Helmholtz equation problem at a corner is derived by means of the contour approach around the singularity. It is discovered that employing the contour approach the jump term comes half and half from the free terms in the L and M kernel integrations, respectively, which differs from the limiting process from an interior point to a boundary point where the jump term is descended from the L kernel only. Thus, the definition of the Hadamard principal value for hypersingular integration at the collocation point of a corner is extended to a generalized sense for both the tangent and normal derivative of double layer potentials as compared to the conventional definition. The free terms of the six kernel functions in the dual integral equations for the Helmholtz equation at a corner have been examined. The kernel functions of the Helmholtz equation are quite different from those of the Laplace equation while the free terms of the Helmholtz equation are the same as those of the Laplace equation. It is worth to point out that the Laplace equation is a special case of the Helmholtz equation when the wave number approaches zero.

Simulation of active noise control using transform domain adaptive filter algorithms linked with the inhomogeneous Helmholtz equation

General three-dimensional solutions are given for the Laplace equation and Helmholtz equation in the cylindrical co-ordinate system, for processes which remain harmonic with time. The expressions for solution which are presented, not only for the magnetic and electrical field variables, but also for the higher-order vector potential, permit an analytical solution. The interrelationships between the field variables, the vector potential, and the higher-order vector potential are discussed. Several examples of engineering applications of the analytical calculation method are given.

Over the last 5 years, classical and optimized Schwarz methods with Robin transmission conditions have been developed for anisotropic elliptic problems discretized by Discrete Duality Finite Volume (DDFV) schemes. We present here the case of higher order transmission conditions in the framework of DDFV. We prove convergence of the algorithm for a large class of symmetric transmission operators, including the discrete Ventcell operator. We also illustrate numerically that using optimized Ventcell transmission conditions leads to much faster algorithms than when using Robin transmission conditions, especially in case of anisotropic elliptic operators.

Objectives: The study reviews the iterative solvers for solving Helmholtz equation. It also surveys the preconditioners for Krylov subspace methods for discrete Helmholtz equation. Numerical experiments based conclusions are drawn, after a thorough comparison of various preconditioners. Methods/Statistical Analysis: Five point finite difference scheme for two-dimensional Helmholtz equation is employed to obtain discrete analogue of Helmholtz equation. Krylov subspace method for is considered for iterations. Matrix based preconditioner ILU with variety of fill-in and operator based preconditioner complex shifted Laplace preconditioner (CSLP) are tested. Different shifts are proposed and experimented in CSLP. Optimization of shifts is recommended on basis of results. Spectral analysis is performed to validate results. Findings: For small wave number problem, ILU type preconditioned performs better and is fairly comparable with CSLP. However, increasing wave number makes unaffordable in terms of computational cost. The pure imaginary shift (0,1) and complex shift (1, / 4 π ) costsless time and number of iterations compared to other choices of shift. These choices are recommended when preconditioner is inverted directly. Approximate inversion may affect iterations, but will save solve time for large problems. Extensive spectral analysis, validating results, highlights this work. Application/Improvements: Dependency upon wave number is decreased with this recommended choice of shifts in CSLP. The number of iterations increases at very low rate when wave number is increased at rate of 2. Keywords: Helmholtz Equation, Iterative Methods, Krylov Subspace Methods, Preconditioners, Spectral Analysis

Heterogeneity can be accounted for by a random potential in the wave equation. For acoustic waves in a fluid with fluctuations of both density and compressibility (as well as for electromagnetic waves in a medium with fluctuation of both permittivity and permeability) the random potential entails a scalar and an operator contribution. For simplicity, the latter is usually overlooked in multiple scattering theory: whatever the type of waves, this simplification amounts to considering the Helmholtz equation with a sound speed c depending on position r. In this work, a radiative transfer equation is derived from the wave equation, in order to study energy transport through a multiple scattering medium. In particular, the influence of the operator term on various transport parameters is studied, based on the diagrammatic approach of multiple scattering. Analytical results are obtained for fundamental quantities of transport theory such as the transport mean-free path ℓ^{*}, scattering phase function f, and anisotropy factor g. Discarding the operator term in the wave equation is shown to have a significant impact on f and g, yet limited to the low-frequency regime, i.e., when the correlation length of the disorder ℓ_{c} is smaller than or comparable to the wavelength λ. More surprisingly, discarding the operator part has a significant impact on the transport mean-free path ℓ^{*} whatever the frequency regime. When the scalar and operator terms have identical amplitudes, the discrepancy on the transport mean-free path is around 300% in the low-frequency regime, and still above 30% for ℓ_{c}/λ=10^{3} no matter how weak fluctuations of the disorder are. Analytical results are supported by numerical simulations of the wave equation and Monte Carlo simulations.

After the successful applications of the combination of the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunctions expansion method (EEM) to solve 2D homogeneous and nonhomogeneous diffusion equations by Young et al. (Young et al., Numer Meth Part Differ Equat 22 (2006), 1173), this article intends to extend the same fundamental concepts to calculate more challenging 3D nonhomogeneous diffusion equations. The nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions are analyzed by the proposed meshless MFS‐MPS‐EEM model. Nonhomogeneous diffusion equation in any complex domains can be decomposed into a Poisson equation and a homogeneous diffusion equation by the principle of linear superposition. This approach is proved to be far better off than solutions by using classic method of separation of variables with inefficient multisummation of very sophisticated series expansion from special functions, which can only limit to treat very simple 3D geometries such as cube, cylinder, or sphere. Poisson equation is solved by using the MPS‐MFS model, in which the source term in the Poisson equation is first handled by the MPS based on the compactly‐supported radial basis functions and the Laplace equation is solved by the MFS. On the other hand, by utilizing the EEM, the homogeneous diffusion equation is first transformed into a Helmholtz equation, which is then solved by the MFS together with the technique of singular value decomposition (SVD) to acquire the eigenvalues and eigenfunctions. After the eigenfunctions are obtained, we can synthesize the diffusion solutions like the orthogonal Fourier series expansions but with only one summation for the series even for multidimensional problems. Numerical results for four case studies of 3D homogeneous and nonhomogeneous diffusion problems show good agreement with the analytical and other numerical solutions, such as finite element method (FEM). Thus, the present numerical scheme has provided a promising meshfree numerical approach to solve 3D nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for very irregular domains. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

A classical result in potential theory is the Schwarz reflection principle for solutions of Laplace's equation which vanish on a portion of a spherical boundary. The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ([4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius. As an application of this result we shall study the problem of the analytic continuation of solutions to the Helmholtz equation defined in the exterior of a bounded domain in three-dimensional Euclidean space ℝ3 We shall show that through the use of the reflection principle derived in this paper, this problem can be reduced to the problem of the analytic continuation of an analytic function of two complex variables, which in turn can be performed through a variety of known methods (cf. [7]).

A simple model is developed, based on the diffusion approximation to the linear transport equation, which gives analytic expressions that describe the coherent propagation of sinusoidally intensity-modulated light through a strongly scattering, low absorbing, homogeneous, infinite medium in terms of the interaction coefficients of the medium. Previously, the analytical multiform solutions to Laplace's equation and the Helmholtz equation were derived respectively by Sommerfeld and Carslaw in their studies of electrostatics, sound, and heat to model intensity modulated light in the presence of an absorbing or reflecting `semi-infinite' plane bounded by a straight edge that is immersed in an infinite, strongly scattering, low absorbing, homogeneous medium. The model predictions are in good agreement with the present results of experiments performed on media consisting of IntralipidTM and skim milk emulsions containing minute quantities of black India ink and with Monte Carlo simulations. These studies provide a theoretical basis for the understanding of photon diffusion in tissues and allow the determination of conditions to obtain maximum resolution and penetration.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

ABSTRACTIn this paper, we give an integral representation of the solution of the Dirichlet problem for the Laplace and Helmholtz equations in the form of a generalized convolution.