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Discontinuous Coefficients (discontinuous + coefficient)
Selected AbstractsComputational methods for optical molecular imagingINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2009Duan Chen Abstract A new computational technique, the matched interface and boundary (MIB) method, is presented to model the photon propagation in biological tissue for the optical molecular imaging. Optical properties have significant differences in different organs of small animals, resulting in discontinuous coefficients in the diffusion equation model. Complex organ shape of small animal induces singularities of the geometric model as well. The MIB method is designed as a dimension splitting approach to decompose a multidimensional interface problem into one-dimensional ones. The methodology simplifies the topological relation near an interface and is able to handle discontinuous coefficients and complex interfaces with geometric singularities. In the present MIB method, both the interface jump condition and the photon flux jump conditions are rigorously enforced at the interface location by using only the lowest-order jump conditions. This solution near the interface is smoothly extended across the interface so that central finite difference schemes can be employed without the loss of accuracy. A wide range of numerical experiments are carried out to validate the proposed MIB method. The second-order convergence is maintained in all benchmark problems. The fourth-order convergence is also demonstrated for some three-dimensional problems. The robustness of the proposed method over the variable strength of the linear term of the diffusion equation is also examined. The performance of the present approach is compared with that of the standard finite element method. The numerical study indicates that the proposed method is a potentially efficient and robust approach for the optical molecular imaging. Copyright © 2008 John Wiley & Sons, Ltd. [source] Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximations of linear elasticity with and without discontinuous coefficientsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2004Andrew C. Bauer Abstract Adaptive finite element methods (FEM) generate linear equation systems that require dynamic and irregular patterns of storage, access, and computation, making their parallelization difficult. Additional difficulties are generated for problems in which the coefficients of the governing partial differential equations have large discontinuities. We describe in this paper the development of a set of iterative substructuring based solvers and domain decomposition preconditioners with an algebraic coarse-grid component that address these difficulties for adaptive hp approximations of linear elasticity with both homogeneous and inhomogeneous material properties. Our solvers are robust and efficient and place no restrictions on the mesh or partitioning. Copyright © 2003 John Wiley & Sons, Ltd. [source] Some results on the accuracy of an edge-based finite volume formulation for the solution of elliptic problems in non-homogeneous and non-isotropic mediaINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2009Darlan Karlo Elisiário de Carvalho Abstract The numerical simulation of elliptic type problems in strongly heterogeneous and anisotropic media represents a great challenge from mathematical and numerical point of views. The simulation of flows in non-homogeneous and non-isotropic porous media with full tensor diffusion coefficients, which is a common situation associated with the miscible displacement of contaminants in aquifers and the immiscible and incompressible two-phase flow of oil and water in petroleum reservoirs, involves the numerical solution of an elliptic type equation in which the diffusion coefficient can be discontinuous, varying orders of magnitude within short distances. In the present work, we present a vertex-centered edge-based finite volume method (EBFV) with median dual control volumes built over a primal mesh. This formulation is capable of handling the heterogeneous and anisotropic media using structured or unstructured, triangular or quadrilateral meshes. In the EBFV method, the discretization of the diffusion term is performed using a node-centered discretization implemented in two loops over the edges of the primary mesh. This formulation guarantees local conservation for problems with discontinuous coefficients, keeping second-order accuracy for smooth solutions on general triangular and orthogonal quadrilateral meshes. In order to show the convergence behavior of the proposed EBFV procedure, we solve three benchmark problems including full tensor, material heterogeneity and distributed source terms. For these three examples, numerical results compare favorably with others found in literature. A fourth problem, with highly non-smooth solution, has been included showing that the EBFV needs further improvement to formally guarantee monotonic solutions in such cases. Copyright © 2008 John Wiley & Sons, Ltd. [source] Convergence of MPFA on triangulations and for Richards' equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2008R. A. Klausen Abstract Spatial discretization of transport and transformation processes in porous media requires techniques that handle general geometry, discontinuous coefficients and are locally mass conservative. Multi-point flux approximation (MPFA) methods are such techniques, and we will here discuss some formulations on triangular grids with further application to the nonlinear Richards equation. The MPFA methods will be rewritten to mixed form to derive stability conditions and error estimates. Several MPFA versions will be shown, and the versions will be discussed with respect to convergence, symmetry and robustness when the grids are rough. It will be shown that the behavior may be quite different for challenging cases of skewness and roughness of the simulation grids. Further, we apply the MPFA discretization approach for the Richards equation and derive new error estimates without extra regularity requirements. The analysis will be accompanied by numerical results for grids that are relevant for practical simulation. Copyright © 2008 John Wiley & Sons, Ltd. [source] Well-posedness, smooth dependence and centre manifold reduction for a semilinear hyperbolic system from laser dynamicsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2007Mark Lichtner Abstract We prove existence, uniqueness, regularity and smooth dependence of the weak solution on the initial data for a semilinear, first order, dissipative hyperbolic system with discontinuous coefficients. Such hyperbolic systems have successfully been used to model the dynamics of distributed feedback multisection semiconductor lasers. We show that in a function space of continuous functions the weak solutions generate a smooth skew product semiflow. Using slow fast structure and dissipativity we prove the existence of smooth exponentially attracting invariant centre manifolds for the non-autonomous model. Copyright © 2006 John Wiley & Sons, Ltd. [source] On parallel solution of linear elasticity problems.NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 3 2002Part II: Methods, some computer experiments Abstract This is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coefficients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method. In the first part of the trilogy block- diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg-method, i.e. inner iterations were performed. As preconditioner, we used modified incomplete factorization MIC(0), where possibly the element matrices were modified in order to give M -matrices, i.e. in order to guarantee the existence of the MIC(0) factorization. In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner/outer iterations we also study a variant of the block-diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)-factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system. In the third part of the trilogy, we will focus on the use of higher-order finite elements. Copyright © 2002 John Wiley & Sons, Ltd. [source] Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficientsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2002N. S. Bakhvalov Abstract We prove extension theorems in the norms described by Stokes and Lamé operators for the three-dimensional case with periodic boundary conditions. For the Lamé equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem. Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and fictitious domain methods, and in analysis of finite element methods. We consider an application of established extension theorems to an efficient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coefficients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coefficient problem. Such preconditioner allows the use of well-known fast algorithms for preconditioning. Under some natural assumptions on smoothness and topological properties of subdomains with small coefficients, we prove convergence of the simplest Richardson method uniform in the jump of coefficients. For the Lamé equations, the convergence is also uniform in the incompressible limit. Our preliminary numerical results for two-dimensional diffusion problems show fast convergence uniform in the jump and in the mesh size parameter. Copyright © 2002 John Wiley & Sons, Ltd. [source] Approximation capability of a bilinear immersed finite element spaceNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2008Xiaoming He Abstract This article discusses a bilinear immersed finite element (IFE) space for solving second-order elliptic boundary value problems with discontinuous coefficients (interface problem). This is a nonconforming finite element space and its partition can be independent of the interface. The error estimates for the interpolation of a Sobolev function indicate that this IFE space has the usual approximation capability expected from bilinear polynomials. Numerical examples of the related finite element method are provided. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 [source] An efficient HOC scheme for transient convection-diffusion-reaction equations with discontinuous coefficients and singular source termsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007Rajendra K Ray In this paper, we propose a new methodology for numerically solving one-dimensional (1D) transient convection-diffusion-reaction equations with discontinuous coefficients and singular source terms on nonuniform space grids. This Higher Order Compact (HOC) formulation is at least third order accurate at regular grid points and exactly third order accurate at points just next to the discontinuity. We conduct numerous numerical studies on a number of problems and compare our results with those obtained with immersed interface and other well-known methods. In all cases our formulation is found to produce better results on relatively coarser grids. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | |||||||||||||