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## Elliptic Problems (elliptic + problem)
## Selected Abstracts## Existence of multiple positive solutions for ,,u,,(u/,x,2)=u2*,1+,f(x) MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2002Ting ChengAbstract In this paper, we consider the semilinear elliptic problem where ,,,N (N,3) is a bounded smooth domain such that 0,,, ,>0 is a real parameter, and f(x) is some given function in L,(,) such that f(x),0, f(x),,0 in ,. Some existence results of multiple solutions have been obtained by implicit function theorem, monotone iteration method and Mountain Pass Lemma. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Weighted isoperimetric inequalities on ,n and applications to rearrangements MATHEMATISCHE NACHRICHTEN, Issue 4 2008M. Francesca BettaAbstract We study isoperimetric inequalities for a certain class of probability measures on ,n together with applications to integral inequalities for weighted rearrangements. Furthermore, we compare the solution to a linear elliptic problem with the solution to some "rearranged" problem defined in the domain {x: x1 < , (x2, ,, xn)} with a suitable function , (x2, ,, xn). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Adaptive finite element approximations on nonmatching grids for second-order elliptic problems NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2010Hongsen ChenAbstract In this article we consider a finite element approximation for a model elliptic problem of second order on non-matching grids. This method combines the continuous finite element method with interior penalty discontinuous Galerkin method. As a special case, we develop a finite element method that is continuous on the matching part of the grid and is discontinuous on the nonmatching part. A residual type a posteriori error estimate is derived. Results of numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] ## Algorithms for vector field generation in mass consistent models NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2010Ciro FloresAbstract Diagnostic models in meteorology are based on the fulfillment of some time independent physical constraints as, for instance, mass conservation. A successful method to generate an adjusted wind field, based on mass conservation equation, was proposed by Sasaki and leads to the solution of an elliptic problem for the multiplier. Here we study the problem of generating an adjusted wind field from given horizontal initial velocity data, by two ways. The first one is based on orthogonal projection in Hilbert spaces and leads to the same elliptic problem but with natural boundary conditions for the multiplier. We derive from this approach the so called E,algorithm. An innovative alternative proposal is obtained from a second approach where we consider the saddle,point formulation of the problem,avoiding boundary conditions for the multiplier, and solving this problem by iterative conjugate gradient methods. This leads to an algorithm that we call the CG,algorithm, which is inspired from Glowinsk's approach to solve Stokes,like problems in computational fluid dynamics. Finally, the introduction of new boundary conditions for the multiplier in the elliptic problem generates better adjusted fields than those obtained with the original boundary conditions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] ## Short fibers suspension in steady recirculating flows THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, Issue 3 2002Francisco ChinestaAbstract Numerical modeling of short fiber suspensions flows involves the coupling between motion equations, which definean elliptic problem, and the fluid constitutive equation, which introduces a non-linear advection problem related to the fiber orientation (induced anisotropy). In a previous work these authors have proposed a numerical procedure to determine a steady solution of the fibers orientation in steady recirculating flows, taking into account that neither initial nor boundary conditions are given. This procedure may be used in the numerical simulation of SFRT flows involving recirculating parts as encountered in the simulationof industrial processes, as well as in inverse rheological identification using, for example, rotative rheometric devices. La modélisation numérique des suspensions de fibres courtes implique le couplage entre les équations de mouvement (qui définissent un problème élliptique) et l'equation constitutive qui introduit un problème de transport non linéaire asocié à l'orientation des fibres. Les auteurs ont proposé, dans des travaux précédents, une technique numérique pour le calcul de l'orientation des fibres dans un écoulement stationnaire recirculant pour lequel les conditions aux limites et les conditions initiates ne sont pas connues. Cette technique peut être utilisée dans la simulation d'écoulements de fibres courtes présentant des recirculations, comme c'est le cas dans les écoulements industrielles en contraction ainsi que dans les instruments rhéométriques rotatifs. [source] ## Best domain for an elliptic problem in cartesian coordinates by means of shape-measure ASIAN JOURNAL OF CONTROL, Issue 5 2009Alireza Fakharzadeh JahromiAbstract In (ZAA J. Anal. Appl., Vol. 16, No. 1, pp. 143,155) we introduced a method to determine the optimal domains for elliptic optimal-shape design problems in polar coordinates. However, the same problem in cartesian coordinates, which are more applicable, is found to be much harder, therefore we had to develop a new approach for these designs. Herein, the unknown domain is divided into a fixed and a variable part and the optimal pair of the domain and its optimal control, is characterized in two stages. Firstly, the optimal control for the each given domain is determined by changing the problem into a measure-theoretical one, replacing this with an infinite dimensional linear programming problem and approximating schemes; then the nearly optimal control function is characterized. Therefore a function that offers the optimal value of the objective function for a given domain, is defined. In the second stage, by applying a standard optimization method, the global minimizer pair will be obtained. Some numerical examples are also given. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source] ## A discontinuous Galerkin method for elliptic interface problems with application to electroporation INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2009Grégory Guyomarc'hAbstract We solve elliptic interface problems using a discontinuous Galerkin (DG) method, for which discontinuities in the solution and in its normal derivatives are prescribed on an interface inside the domain. Standard ways to solve interface problems with finite element methods consist in enforcing the prescribed discontinuity of the solution in the finite element space. Here, we show that the DG method provides a natural framework to enforce both discontinuities weakly in the DG formulation, provided the triangulation of the domain is fitted to the interface. The resulting discretization leads to a symmetric system that can be efficiently solved with standard algorithms. The method is shown to be optimally convergent in the L2 -norm. We apply our method to the numerical study of electroporation, a widely used medical technique with applications to gene therapy and cancer treatment. Mathematical models of electroporation involve elliptic problems with dynamic interface conditions. We discretize such problems into a sequence of elliptic interface problems that can be solved by our method. We obtain numerical results that agree with known exact solutions. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2009M. PicassoAbstract We propose a simple stopping criterion for the conjugate gradient (CG) algorithm in the framework of anisotropic, adaptive finite elements for elliptic problems. The goal of the adaptive algorithm is to find a triangulation such that the estimated relative error is close to a given tolerance TOL. We propose to stop the CG algorithm whenever the residual vector has Euclidian norm less than a small fraction of the estimated error. This stopping criterion is based on a posteriori error estimates between the true solution u and the computed solution u (the superscript n stands for the CG iteration number, the subscript h for the typical mesh size) and on heuristics to relate the error between uh and u to the residual vector. Numerical experiments with anisotropic adaptive meshes show that the total number of CG iterations can be divided by 10 without significant discrepancy in the computed results. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## An integral-collocation-based fictitious-domain technique for solving elliptic problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2008N. Mai-DuyAbstract This paper presents a new fictitious-domain technique for numerically solving elliptic second-order partial differential equations (PDEs) in complex geometries. The proposed technique is based on the use of integral-collocation schemes and Chebyshev polynomials. The boundary conditions on the actual boundary are implemented by means of integration constants. The method works for both Dirichlet and Neumann boundary conditions. Several test problems are considered to verify the technique. Numerical results show that the present method yields spectral accuracy for smooth (analytic) problems. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Some superconvergence results for the covolume method for elliptic problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2001Jianguo HuangAbstract In this paper, we attempt to give analysis of the covolume method for solving general self-adjoint elliptic problems. We first present some useful superconvergence results for the deviation between the solution of the covolume method and the solution of the induced finite element method, in the energy norm and maximum norm, respectively. With these results, we then reproduce the maximum norm estimates obtained by Chou and Li for the covolume method easily. Furthermore, based on the covolume method, we propose a high-accuracy algorithm for solving general self-adjoint elliptic problems. Compared with the original covolume method, the computation work of the new algorithm is increased slightly, but the approximate error is improved remarkably. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2010Clark R. DohrmannAbstract Overlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on spaces, with a number of degrees of freedom per subdomain which are uniformly bounded, which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e. methods based on nonoverlapping decompositions of the domain. The local components of the new preconditioners are based on solvers on a set of overlapping subdomains. In the current study, the dimension of the coarse spaces is smaller than in recently developed algorithms; in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed. In many cases, this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered. In addition, in spite of using overlapping subdomains to define the local components of the preconditioner, values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid Schwarz algorithm. The use of discontinuous pressures makes it possible to work exclusively with symmetric, positive-definite problems and the standard preconditioned conjugate gradient method. Bounds are established for the condition number of the preconditioned operators. The bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains, and it is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. Numerical results illustrate the findings. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2008Lourenço Beirão da VeigaAbstract We present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form. We show that this estimator is reliable and efficient with respect to an energy-type error comprising both flux and pressure. Its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes, different permeability tensors, and different types of computational meshes. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Multiscale Galerkin method using interpolation wavelets for two-dimensional elliptic problems in general domains INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2004Gang-Won JangAbstract One major hurdle in developing an efficient wavelet-based numerical method is the difficulty in the treatment of general boundaries bounding two- or three-dimensional domains. The objective of this investigation is to develop an adaptive multiscale wavelet-based numerical method which can handle general boundary conditions along curved boundaries. The multiscale analysis is achieved in a multi-resolution setting by employing hat interpolation wavelets in the frame of a fictitious domain method. No penalty term or the Lagrange multiplier need to be used in the present formulation. The validity of the proposed method and the effectiveness of the multiscale adaptive scheme are demonstrated by numerical examples dealing with the Dirichlet and Neumann boundary-value problems in quadrilateral and quarter circular domains. 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 media INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2009Darlan Karlo Elisiário de CarvalhoAbstract 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] ## 2D thermal/isothermal incompressible viscous flows INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2005Alfredo NicolásAbstract 2D thermal and isothermal time-dependent incompressible viscous flows are presented in rectangular domains governed by the Boussinesq approximation and Navier,Stokes equations in the stream function,vorticity formulation. The results are obtained with a simple numerical scheme based on a fixed point iterative process applied to the non-linear elliptic systems that result after a second-order time discretization. The iterative process leads to the solution of uncoupled, well-conditioned, symmetric linear elliptic problems. Thermal and isothermal examples are associated with the unregularized, driven cavity problem and correspond to several aspect ratios of the cavity. Some results are presented as validation examples and others, to the best of our knowledge, are reported for the first time. The parameters involved in the numerical experiments are the Reynolds number Re, the Grashof number Gr and the aspect ratio. All the results shown correspond to steady state flows obtained from the unsteady problem. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Homogenization of elliptic problems with the Dirichlet and Neumann conditions imposed on varying subsets MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2007Carmen Calvo-JuradoAbstract We study the asymptotic behaviour of the solution un of a linear elliptic equation posed in a fixed domain ,. The solution un is assumed to satisfy a Dirichlet boundary condition on ,n, where ,n is an arbitrary sequence of subsets of ,,, and a Neumman boundary condition on the remainder of ,,. We obtain a representation of the limit problem which is stable by homogenization and where it appears a generalized Fourier boundary condition. We also prove a corrector result. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2004A. Nazarov, SergueAbstract The approximation of solutions to boundary value problems on unbounded domains by those on bounded domains is one of the main applications for artificial boundary conditions. Based on asymptotic analysis, here a new method is presented to construct local artificial boundary conditions for a very general class of elliptic problems where the main asymptotic term is not known explicitly. Existence and uniqueness of approximating solutions are proved together with asymptotically precise error estimates. One class of important examples includes boundary value problems for anisotropic elasticity and piezoelectricity. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## The factorization method for a class of inverse elliptic problems MATHEMATISCHE NACHRICHTEN, Issue 3 2005Andreas KirschAbstract In this paper the factorization method from inverse scattering theory and impedance tomography is extended to a class of general elliptic differential equations in divergence form. The inverse problem is to determine the interface ,, of an interior change of the material parameters from the Neumann-Dirichlet map. Since absorption is allowed a suitable combination of the real and imaginary part of the Neumann-Dirichlet map is needed to explicitely characterize , by the data. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Infinitely many solutions for polyharmonic elliptic problems with broken symmetries MATHEMATISCHE NACHRICHTEN, Issue 1 2003Sergio LancelottiAbstract By means of a perturbation argument devised by P. Bolle, we prove the existence of infinitely many solutions for perturbed symmetric polyharmonic problems with non,homogeneous Dirichlet boundary conditions. An extension to the higher order case of the estimate from below for the critical values due to K. Tanaka is obtained. [source] ## ON AXISYMMETRIC TRAVELING WAVES AND RADIAL SOLUTIONS OF SEMI-LINEAR ELLIPTIC EQUATIONS NATURAL RESOURCE MODELING, Issue 3 2000THOMAS P. WITELSKIABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementaryapproach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowlyvarying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved front solutions. These axisym-metric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensivelydiscuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Manyof the results contained in this article are known, and in presenting known results, it is intended that this article be expositoryin nature, providing elementarydemonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radiallysymmetric solutions of semilinear elliptic problems, especiallyto those articles in which more advanced mathematical theoryis developed. [source] ## Convergence of adaptive edge finite element methods for H(curl)-elliptic problems NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2010Liuqiang ZhongAbstract The standard adaptive edge finite element method (AEFEM), using first/second family Nédélec edge elements with any order, for the three-dimensional H(curl)-elliptic problems with variable coefficients is shown to be convergent for the sum of the energy error and the scaled error estimator. The special treatment of the data oscillation and the interior node property are removed from the proof. Numerical experiments indicate that the adaptive meshes and the associated numerical complexity are quasi-optimal. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## On the multilevel preconditioning of Crouzeix,Raviart elliptic problems NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2008J. KrausAbstract We consider robust hierarchical splittings of finite element spaces related to non-conforming discretizations using Crouzeix,Raviart type elements. As is well known, this is the key to the construction of efficient two- and multilevel preconditioners. The main contribution of this paper is a theoretical and an experimental comparison of three such splittings. Our starting point is the standard method based on differences and aggregates (DA) as introduced in Blaheta et al. (Numer. Linear Algebra Appl. 2004; 11:309,326). On this basis we propose a more general (GDA) splitting, which can be viewed as the solution of a constraint optimization problem (based on certain symmetry assumptions). We further consider the locally optimal (ODA) splitting, which is shown to be equivalent to the first reduce (FR) method from Blaheta et al. (Numer. Linear Algebra Appl. 2004; 11:309,326). This means that both, the ODA and the FR splitting, generate the same subspaces, and thus the local constant in the strengthened Cauchy,Bunyakowski,Schwarz inequality is minimal for the FR (respectively ODA) splitting. Moreover, since the DA splitting corresponds to a particular choice in the parameter space of the GDA splitting, which itself is an element in the set of all splittings for which the ODA (or equivalently FR) splitting yields the optimum, we conclude that the chain of inequalities ,,,,,,3/4 holds independently of mesh and/or coefficient anisotropy. Apart from the theoretical considerations, the presented numerical results provide a basis for a comparison of these three approaches from a practical point of view. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2008J. K. KrausAbstract We construct optimal order multilevel preconditioners for interior-penalty discontinuous Galerkin (DG) finite element discretizations of three-dimensional (3D) anisotropic elliptic boundary-value problems. In this paper, we extend the analysis of our approach, introduced earlier for 2D problems (SIAM J. Sci. Comput., accepted), to cover 3D problems. A specific assembling process is proposed, which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. A new bound for the constant , in the strengthened Cauchy,Bunyakowski,Schwarz inequality is derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Adaptive finite element approximations on nonmatching grids for second-order elliptic problems NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2010Hongsen ChenAbstract In this article we consider a finite element approximation for a model elliptic problem of second order on non-matching grids. This method combines the continuous finite element method with interior penalty discontinuous Galerkin method. As a special case, we develop a finite element method that is continuous on the matching part of the grid and is discontinuous on the nonmatching part. A residual type a posteriori error estimate is derived. Results of numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] ## Radial basis collocation method and quasi-Newton iteration for nonlinear elliptic problems NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2008H.Y. HuAbstract 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] ## A remark on the coercivity for a first-order least-squares method NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2007Jaeun KuAbstract This study present a short proof concerning the coercivity of a first-order least-squares finite element method for general second-order elliptic problems proposed by Cai, Lazarov, Manteuffel and McCormick (Cai et al. J Numer Anal 31 (1994), 1785,1799). Our proof is based on a priori estimate and the technique can be applied to prove L2 -norm error estimate for the primary function u. After establishing the coercivity bound from the assumed a priori estimate, we observe that the coercivity bound is actually equivalent to the a priori estimate. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] ## Discontinuous Galerkin methods for periodic boundary value problems NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2007Kumar VemagantiAbstract This article considers the extension of well-known discontinuous Galerkin (DG) finite element formulations to elliptic problems with periodic boundary conditions. Such problems routinely appear in a number of applications, particularly in homogenization of composite materials. We propose an approach in which the periodicity constraint is incorporated weakly in the variational formulation of the problem. Both H1 and L2 error estimates are presented. A numerical example confirming theoretical estimates is shown. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007 [source] ## A posteriori error estimator for expanded mixed hybrid methods, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2007Dongho KimAbstract In this article, we construct an a posteriori error estimator for expanded mixed hybrid finite-element methods for second-order elliptic problems. An a posteriori error analysis yields reliable and efficient estimate based on residuals. Several numerical examples are presented to show the effectivity of our error indicators. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 330,349, 2007 [source] ## Additive Schwarz-type preconditioners for fourth-order elliptic problems using Hermite cubic splines NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2006Song-Tao LiuAbstract In this note, we construct the additive Schwarz-type preconditioner for fourth-order elliptic problems with nested subspaces from Hermite cubic splines. We prove that after the preconditioning, the system is well conditioned. The numerical evidence strongly supports our theoretical result. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source] ## A cell boundary element method for elliptic problems, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2005Youngmok JeonAbstract An elementary analysis on the cell boundary element (CBEM) was given by Jeon and Sheen. In this article we improve the previous results in various aspects. First of all, stability and convergence analysis on the rectangular grids are established. Moreover, error estimates are improved. Our improved analysis was possible by recasting of the CBEM in a Petrov-Galerkin setting. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] |