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Elliptic Partial Differential Equations (elliptic + partial_differential_equation)
Selected AbstractsTorsion of orthotropic bars with L -shaped or cruciform cross-sectionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2001Y. Z. Chen Abstract For an orthotropic torsion bar with L -shaped or cruciform cross-section, the studied torsion problem can be reduced to a boundary value problem of elliptic partial differential equation. The studied region is separated into several rectangular sub-regions, and the series solution is suggested to solve the problem for the individual sub-region. By using the continuation condition for the functions on the neighbouring sub-regions, the investigated solution is obtainable. Finally, numerical results for the torsion rigidities of bars are given to demonstrate the influence of the degree of orthotropy. Copyright © 2001 John Wiley & Sons, Ltd. [source] Local discretization error bounds using interval boundary element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009B. F. Zalewski Abstract In this paper, a method to account for the point-wise discretization error in the solution for boundary element method is developed. Interval methods are used to enclose the boundary integral equation and a sharp parametric solver for the interval linear system of equations is presented. The developed method does not assume any special properties besides the Laplace equation being a linear elliptic partial differential equation whose Green's function for an isotropic media is known. Numerical results are presented showing the guarantee of the bounds on the solution as well as the convergence of the discretization error. Copyright © 2008 John Wiley & Sons, Ltd. [source] On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level SchwarzINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 6 2002R. S. Tuminaro Abstract Multilevel methods offer the best promise to attain both fast convergence and parallel efficiency in the numerical solution of parabolic and elliptic partial differential equations. Unfortunately, they have not been widely used in part because of implementation difficulties for unstructured mesh solvers. To facilitate use, a multilevel preconditioner software module, ML, has been constructed. Several methods are provided requiring relatively modest programming effort on the part of the application developer. This report discusses the implementation of one method in the module: a two-level Krylov,Schwarz preconditioner. To illustrate the use of these methods in computational fluid dynamics (CFD) engineering applications, we present results for 2D and 3D CFD benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd. [source] From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2004Anis Younes Abstract The link between Mixed Finite Element (MFE) and Finite Volume (FV) methods applied to elliptic partial differential equations has been investigated by many authors. Recently, a FV formulation of the mixed approach has been developed. This approach was restricted to 2D problems with a scalar for the parameter used to calculate fluxes from the state variable gradient. This new approach is extended to 2D problems with a full parameter tensor and to 3D problems. The objective of this new formulation is to reduce the total number of unknowns while keeping the same accuracy. This is achieved by defining one new variable per element. For the 2D case with full parameter tensor, this new formulation exists for any kind of triangulation. It allows the reduction of the number of unknowns to the number of elements instead of the number of edges. No additional assumptions are required concerning the averaging of the parameter in hetero- geneous domains. For 3D problems, we demonstrate that the new formulation cannot exist for a general 3D tetrahedral discretization, unlike in the 2D problem. However, it does exist when the tetrahedrons are regular, or deduced from rectangular parallelepipeds, and allows reduction of the number of unknowns. Numerical experiments and comparisons between both formulations in 2D show the efficiency of the new formulation. Copyright © 2003 John Wiley & Sons, Ltd. [source] Analysis of parameter sensitivity and experimental design for a class of nonlinear partial differential equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2005Michael L. Anderson Abstract The purpose of this work is to analyse the parameter sensitivity problem for a class of nonlinear elliptic partial differential equations, and to show how numerical simulations can help to optimize experiments for the estimation of parameters in such equations. As a representative example we consider the Laplace,Young problem describing the free surface between two fluids in contact with the walls of a bounded domain, with the parameters being those associated with surface tension and contact. We investigate the sensitivity of the solution and associated functionals to the parameters, examining in particular under what conditions the solution is sensitive to parameter choice. From this, the important practical question of how to optimally design experiments is discussed; i.e. how to choose the shape of the domain and the type of measurements to be performed, such that a subsequent inversion of the measured data for the model parameters yields maximal accuracy in the parameters. We investigate this through numerical studies of the behaviour of the eigenvalues of the sensitivity matrix and their relation to experimental design. These studies show that the accuracy with which parameters can be identified from given measurements can be improved significantly by numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd. [source] Large eddy simulation of turbulent flows via domain decomposition techniques.INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2005Part 2: applications Abstract The present paper discusses the application of large eddy simulation to incompressible turbulent flows in complex geometries. Algorithmic developments concerning the flow solver were provided in the companion paper (Int. J. Numer. Meth. Fluids, 2003; submitted), which addressed the development and validation of a multi-domain kernel suitable for the integration of the elliptic partial differential equations arising from the fractional step procedure applied to the incompressible Navier,Stokes equations. Numerical results for several test problems are compared to reference experimental and numerical data to demonstrate the potential of the method. Copyright © 2005 John Wiley & Sons, Ltd. [source] OPTIMAL HARVESTING OF A SPATIALLY EXPLICIT FISHERY MODELNATURAL RESOURCE MODELING, Issue 2 2009WANDI DING Abstract We consider an optimal fishery harvesting problem using a spatially explicit model with a semilinear elliptic PDE, Dirichlet boundary conditions, and logistic population growth. We consider two objective functionals: maximizing the yield and minimizing the cost or the variation in the fishing effort (control). Existence, necessary conditions, and uniqueness for the optimal harvesting control for both cases are established. Results for maximizing the yield with Neumann (no-flux) boundary conditions are also given. The optimal control when minimizing the variation is characterized by a variational inequality instead of the usual algebraic characterization, which involves the solutions of an optimality system of nonlinear elliptic partial differential equations. Numerical examples are given to illustrate the results. [source] An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matricesNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2010M. Donatelli Abstract Local Fourier analysis (LFA) is a classical tool for proving convergence theorems for multigrid methods (MGMs). In particular, we are interested in optimal convergence, i.e. convergence rates that are independent of the problem size. For elliptic partial differential equations (PDEs), a well-known optimality result requires that the sum of the orders of the grid transfer operators is not lower than the order of the PDE approximated. Analogously, when dealing with MGMs for Toeplitz matrices, a well-known optimality condition concerns the position and the order of the zeros of the symbols of the grid transfer operators. In this work we show that in the case of elliptic PDEs with constant coefficients, the two different approaches lead to an equivalent condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA, which allows to deal not only with differential problems but also for instance with integral problems. The equivalence of the two approaches gives the possibility of using grid transfer operators with different orders also for MGMs for Toeplitz matrices. We give also a class of grid transfer operators related to the B-spline's refinement equation and study their geometric properties. Numerical experiments confirm the correctness of the proposed analysis. Copyright © 2010 John Wiley & Sons, Ltd. [source] IBLU decompositions based on Padé approximantsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 8 2008A. Buzdin Abstract In this paper, we introduce a new class of frequency-filtering IBLU decompositions that use continued-fraction approximation for the diagonal blocks. This technique allows us to construct efficient frequency-filtering preconditioners for discretizations of elliptic partial differential equations on domains with non-trivial geometries. We prove theoretically for a class of model problems that the application of the proposed preconditioners leads to a convergence rate of up to 1,O(h1/4) of the CG iteration. Copyright © 2008 John Wiley & Sons, Ltd. [source] On some versions of the element agglomeration AMGe methodNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2008Ilya Lashuk Abstract The present paper deals with element-based algebraic multigrid (AMGe) methods that target linear systems of equations coming from finite element discretizations of elliptic partial differential equations. The individual element information (element matrices and element topology) is the main input to construct the AMG hierarchy. We study a number of variants of the spectral agglomerate AMGe method. The core of the algorithms relies on element agglomeration utilizing the element topology (built recursively from fine to coarse levels). The actual selection of the coarse degrees of freedom is based on solving a large number of local eigenvalue problems. Additionally, we investigate strategies for adaptive AMG as well as multigrid cycles that are more expensive than the V -cycle utilizing simple interpolation matrices and nested conjugate gradient (CG)-based recursive calls between the levels. The presented algorithms are illustrated with an extensive set of experiments based on a matlab implementation of the methods. Copyright © 2008 John Wiley & Sons, Ltd. [source] Robust parameter-free algebraic multilevel preconditioningNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6-7 2002Y. Notay Abstract To precondition large sparse linear systems resulting from the discretization of second-order elliptic partial differential equations, many recent works focus on the so-called algebraic multilevel methods. These are based on a block incomplete factorization process applied to the system matrix partitioned in hierarchical form. They have been shown to be both robust and efficient in several circumstances, leading to iterative solution schemes of optimal order of computational complexity. Now, despite the procedure is essentially algebraic, previous works focus generally on a specific context and consider schemes that use classical grid hierarchies with characteristic mesh sizes h,2h,4h, etc. Therefore, these methods require some extra information besides the matrix of the linear system and lack of robustness in some situations where semi-coarsening would be desirable. In this paper, we develop a general method that can be applied in a black box fashion to a wide class of problems, ranging from 2D model Poisson problems to 3D singularly perturbed convection,diffusion equations. It is based on an automatic coarsening process similar to the one used in the AMG method, and on coarse grid matrices computed according to a simple and cheap aggregation principle. Numerical experiments illustrate the efficiency and the robustness of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd. [source] Radial basis collocation method and quasi-Newton iteration for nonlinear elliptic problemsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2008H.Y. Hu Abstract This work presents a radial basis collocation method combined with the quasi-Newton iteration method for solving semilinear elliptic partial differential equations. The main result in this study is that there exists an exponential convergence rate in the radial basis collocation discretization and a superlinear convergence rate in the quasi-Newton iteration of the nonlinear partial differential equations. In this work, the numerical error associated with the employed quadrature rule is considered. It is shown that the errors in Sobolev norms for linear elliptic partial differential equations using radial basis collocation method are bounded by the truncation error of the RBF. The combined errors due to radial basis approximation, quadrature rules, and quasi-Newton and Newton iterations are also presented. This result can be extended to finite element or finite difference method combined with any iteration methods discussed in this work. The numerical example demonstrates a good agreement between numerical results and analytical predictions. The numerical results also show that although the convergence rate of order 1.62 of the quasi-Newton iteration scheme is slightly slower than rate of order 2 in the Newton iteration scheme, the former is more stable and less sensitive to the initial guess. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] | |||||||||||||