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Poisson Problems (poisson + problem)
Selected AbstractsRadial basis functions for solving near singular Poisson problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2003C. S. Chen Abstract In this paper, we investigate the use of radial basis functions for solving Poisson problems with a near-singular inhomogeneous source term. The solution of the Poisson problem is first split into two parts: near-singular solution and smooth solution. A method for evaluating the near-singular particular solution is examined. The smooth solution is further split into a particular solution and a homogeneous solution. The MPS-DRM approach is adopted to evaluate the smooth solution. Copyright © 2003 John Wiley & Sons, Ltd. [source] On the mixed finite element method with Lagrange multipliersNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003Ivo Babu Abstract In this note we analyze a modified mixed finite element method for second-order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R2. The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu,ka-Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart-Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192,210, 2003 [source] Local defect correction with different grid typesNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2002V. Nefedov Abstract For a Poisson problem with a solution having large gradients in (nearly) circular subregions a local defect correction method is considered. The problem on the global domain is discretized on a cartesian grid, whereas the restriction of the problem to a circular subdomain is discretized on a polar grid. The two discretizations are then combined in an iterative way. We show that LDC can be viewed as an iterative method for the Poisson equation on a single composite cartesian-polar grid. The efficiency of methods is illustrated by numerical examples. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 454,468, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10018 [source] Collocation methods based on radial basis functions for solving stochastic Poisson problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2007Somchart ChantasiriwanArticle first published online: 19 JUN 200 Abstract Collocation methods based on radial basis functions can be used to provide accurate solutions to deterministic problems. For stochastic problems, accurate solutions may not be desirable if they are too sensitive to random inputs. In this paper, four methods are used to solve stochastic Poisson problems by expressing solutions in terms of source terms and boundary conditions. Comparison among the methods reveals that the method based on fundamental solutions performs better than other methods. Copyright © 2006 John Wiley & Sons, Ltd. [source] Solution of two-dimensional Poisson problems in quadrilateral domains using transfinite Coons interpolationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2004Christopher G. Provatidis Abstract This paper proposes a global approximation method to solve elliptic boundary value Poisson problems in arbitrary shaped 2-D domains. Using transfinite interpolation, a symmetric finite element formulation is derived for degrees of freedom arranged mostly along the boundary of the domain. In cases where both Dirichlet and Neumann boundary conditions occur, the numerical solution is based on bivariate Coons interpolation using the boundary only. Furthermore, in case of only Dirichlet boundary conditions and no existing axes of symmetry, it is proposed to use at least one internal point and apply transfinite interpolation. The theory is sustained by five numerical examples applied to domains of square, circular and elliptic shape. Copyright © 2004 John Wiley & Sons, Ltd. [source] Application of the additive Schwarz method to large scale Poisson problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2004K. M. Singh Abstract This paper presents an application of the additive Schwarz method to large scale Poisson problems on parallel computers. Domain decomposition in rectangular blocks with matching grids on a structured rectangular mesh has been used together with a stepwise approximation to approximate sloping sides and complicated geometric features. A seven-point stencil based on central difference scheme has been used for the discretization of the Laplacian for both interior and boundary grid points, and this results in a symmetric linear algebraic system for any type of boundary conditions. The preconditioned conjugate gradient method has been used as an accelerator for the additive Schwarz method, and three different methods have been assessed for the solution of subdomain problems. Numerical experiments have been performed to determine the most suitable set of subdomain solvers and the optimal accuracy of subdomain solutions; to assess the effect of different decompositions of the problem domain; and to evaluate the parallel performance of the additive Schwarz preconditioner. Application to a practical problem involving complicated geometry is presented which establishes the efficiency and robustness of the method. Copyright © 2004 John Wiley & Sons, Ltd. [source] Radial basis functions for solving near singular Poisson problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2003C. S. Chen Abstract In this paper, we investigate the use of radial basis functions for solving Poisson problems with a near-singular inhomogeneous source term. The solution of the Poisson problem is first split into two parts: near-singular solution and smooth solution. A method for evaluating the near-singular particular solution is examined. The smooth solution is further split into a particular solution and a homogeneous solution. The MPS-DRM approach is adopted to evaluate the smooth solution. Copyright © 2003 John Wiley & Sons, Ltd. [source] Boundary value problems for Dirac operators and Maxwell's equations in non-smooth domainsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16-18 2002Marius Mitrea Abstract We study the well-posedness of the half-Dirichlet and Poisson problems for Dirac operators in three-dimensional Lipschitz domains, with a special emphasis on optimal Lebesgue and Sobolev-Besov estimates. As an application, an elliptization procedure for the Maxwell system is devised. Copyright © 2002 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] | ||||||||||