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Parabolic Problems (parabolic + problem)
Selected AbstractsAnisotropic mesh adaption for time-dependent problems,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2008S. Micheletti Abstract We propose a space,time adaptive procedure for a model parabolic problem based on a theoretically sound anisotropic a posteriori error analysis. A space,time finite element scheme (continuous in space but discontinuous in time) is employed to discretize this problem, thus allowing for non-matching meshes at different time levels. Copyright © 2008 John Wiley & Sons, Ltd. [source] Asymptotic behaviour of solutions of quasilinear evolutionary partial differential equations of parabolic type on unbounded spatial intervalsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2006Poul Abstract We study the long-time behaviour of solutions to a quasilinear parabolic problem on a half-line. The main result lies in showing the existence of a positive solution that converges to the travelling wave of solution to the stationary problem on the whole line. The main tools used here are the zero number theory and the concentration compactness principle. This result is a generalization of a result know for semilinear parabolic equations. Copyright © 2006 John Wiley & Sons, Ltd. [source] Semilinear parabolic problem with nonstandard boundary conditions: Error estimatesNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003Marián Slodi Abstract We study a semilinear parabolic partial differential equation of second order in a bounded domain , , ,N, with nonstandard boundary conditions (BCs) on a part ,non of the boundary ,,. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux through ,non is given, and the solution along ,non has to follow a prescribed shape function, apart from an additive (unknown) space-constant ,(t). We prove the well-posedness of the problem, provide a numerical method for the recovery of the unknown boundary data, and establish the error estimates. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 167,191, 2003 [source] A fractional splitting algorithm for nonoverlapping domain decomposition for parabolic problemNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2002Daoud S. Daoud Abstract In this article we study the convergence of the nonoverlapping domain decomposition for solving large linear system arising from semi-discretization of two-dimensional initial value problem with homogeneous boundary conditions and solved by implicit time stepping using first and two alternatives of second-order FS-methods. The interface values along the artificial boundary condition line are found using explicit forward Euler's method for the first-order FS-method, and for the second-order FS-method to use extrapolation procedure for each spatial variable individually. The solution by the nonoverlapping domain decomposition with FS-method is applicable to problems that requires the solution on nonuniform meshes for each spatial variable, which will enable us to use different time-stepping over different subdomains and with the possibility of extension to three-dimensional problem. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 609,624, 2002 [source] Discontinuous Galerkin framework for adaptive solution of parabolic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2007Deepak V. Kulkarni Abstract Non-conforming meshes are frequently employed in adaptive analyses and simulations of multi-component systems. We develop a discontinuous Galerkin formulation for the discretization of parabolic problems that weakly enforces continuity across non-conforming mesh interfaces. A benefit of the DG scheme is that it does not introduce constraint equations and their resulting Lagrange multiplier fields as done in mixed and mortar methods. The salient features of the formulation are highlighted through an a priori analysis. When coupled with a mesh refinement scheme the DG formulation is able to accommodate multiple hanging nodes per element edge and leads to an effective adaptive framework for the analysis of interface evolution problems. We demonstrate our approach by analysing the Stefan problem of solidification. Copyright © 2006 John Wiley & Sons, Ltd. [source] A review of reliable numerical models for three-dimensional linear parabolic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2007I. Faragó Abstract The preservation of characteristic qualitative properties of different phenomena is a more and more important requirement in the construction of reliable numerical models. For phenomena that can be mathematically described by linear partial differential equations of parabolic type (such as the heat conduction, the diffusion, the pricing of options, etc.), the most important qualitative properties are: the maximum,minimum principle, the non-negativity preservation and the maximum norm contractivity. In this paper, we analyse the discrete analogues of the above properties for finite difference and finite element models, and we give a systematic overview of conditions that guarantee the required properties a priori. We have chosen the heat conduction process to illustrate the main concepts, but engineers and scientists involved in scientific computing can easily reformulate the results for other problems too. Copyright © 2006 John Wiley & Sons, Ltd. [source] Guaranteed a posteriori error estimation for fully discrete solutions of parabolic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2003T. Strouboulis Abstract This paper addresses the construction of guaranteed computable estimates for fully discrete solutions of parabolic problems. Copyright © 2003 John Wiley & Sons, Ltd. [source] The maximum principle violations of the mixed-hybrid finite-element method applied to diffusion equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2002H. Hoteit Abstract The abundant literature of finite-element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial,boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time-dependent solutions for such problems during small time duration obtained by using a non-conforming mixed-hybrid finite-element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite-difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non-physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial,boundary value problem will be presented. One of the propositions given to avoid any non-physical oscillations is to use the mass-lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd. [source] Blow up, decay bounds and continuous dependence inequalities for a class of quasilinear parabolic problemsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2006L. E. Payne Abstract This paper deals with a class of semilinear parabolic problems. In particular, we establish conditions on the data sufficient to guarantee blow up of solution at some finite time, as well as conditions which will insure that the solution exists for all time with exponential decay of the solution and its spatial derivatives. In the case of global existence, we also investigate the continuous dependence of the solution with respect to some data of the problem. Copyright © 2005 John Wiley & Sons, Ltd. [source] Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problemsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2006G. A. Philippin Abstract This paper deals with a class of semilinear parabolic problems. We establish sufficient conditions on the data forcing the solution to blow up at finite time , and derive an upper bound for ,. Moreover, we show that if the problem is modified in some way, the solution decays exponentially in time and depends continuously on the data. Copyright © 2005 John Wiley & Sons, Ltd. [source] Elliptic and parabolic problems in unbounded domainsMATHEMATISCHE NACHRICHTEN, Issue 1 2004Patrick Guidotti Abstract We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi-explicit representation formulas via operator-valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] High order smoothing schemes for inhomogeneous parabolic problems with applications in option pricingNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2007A.Q.M. Khaliq Abstract A new family of numerical schemes for inhomogeneous parabolic partial differential equations is developed utilizing diagonal Padé schemes combined with positivity,preserving Padé schemes as damping devices. We also develop a split version of the algorithm using partial fraction decomposition to address difficulties with accuracy and computational efficiency in solving and to implement the algorithms in parallel. Numerical experiments are presented for several inhomogeneous parabolic problems, including pricing of financial options with nonsmooth payoffs.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] Lp error estimates and superconvergence for covolume or finite volume element methodsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2003So-Hsiang Chou Abstract We consider convergence of the covolume or finite volume element solution to linear elliptic and parabolic problems. Error estimates and superconvergence results in the Lp norm, 2 , p , ,, are derived. We also show second-order convergence in the Lp norm between the covolume and the corresponding finite element solutions and between their gradients. The main tools used in this article are an extension of the "supercloseness" results in Chou and Li [Math Comp 69(229) (2000), 103,120] to the Lp based spaces, duality arguments, and the discrete Green's function method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 463,486, 2003 [source] | |||||||||||||