Element Method For Parabolic Problems Research Articles

A quadratic finite volume element method for parabolic problems on quadrilateral meshes

In this paper we utilize affine biquadratic elements and a two-step temporal discretization to develop a finite volume element method for parabolic problems on quadrilateral meshes. The method is proved to have an optimal order convergence rate in L2(0, T; H1(Ω)) under the ‘asymptotically parallelogram’ mesh assumption. Numerical experiments that corroborate the theoretical analysis are also presented.

Superconvergence of H 1-Galerkin mixed finite element methods for parabolic problems

In this article, we study the semidiscrete H 1-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L 2-norm for the flux is of order 𝒪(h k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h k+3) between the H 1-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h k+3) is realized via a postprocessing technique using local projection methods.

Discontinuous finite volume element method for parabolic problems

Discontinuous finite volume element method for parabolic problems

Mortar element methods for parabolic problems

AbstractIn this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L∞(L2) and L∞(H1)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

A discontinuous Galerkin Method for parabolic problems with modified hp-finite element approximation technique

A recent paper [Hideaki Kaneko, Kim S. Bey, Gene J.W. Hou, Discontinuous Galerkin finite element method for parabolic problems, preprint November 2000, NASA] is generalized to a case where the spatial region is taken in R 3. The region is assumed to be a thin body, such as a panel on the wing or fuselage of an aerospace vehicle. The traditional h- as well as hp-finite element methods are applied to the surface defined in the x– y variables, while, through the thickness, the technique of the p-element is employed. Time and spatial discretization scheme developed in Kaneko et al. (2000), based upon an assumption of certain weak singularity of ∥ u t ∥ 2, is used to derive an optimal a priori error estimate for the current method.

Spectral element methods for parabolic problems

A spectral element method for solving parabolic initial boundary value problems on smooth domains using parallel computers is presented in this paper. The space domain is divided into a number of shape regular quadrilaterals of size h and the time step k is proportional to h 2 . At each time step we minimize a functional which is the sum of the squares of the residuals in the partial differential equation, initial condition and boundary condition in different Sobolev norms and a term which measures the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The Sobolev spaces used are of different orders in space and time. We can define a preconditioner for the minimization problem which allows the problem to decouple. Error estimates are obtained for both the h and p versions of this method.

Discontinuous Galerkin finite element method for parabolic problems

In this paper, we develop a time and its corresponding spatial discretization scheme, based upon the assumption of a certain weak singularity of ‖ u t ( t ) ‖ L 2 ( Ω ) = ‖ u t ‖ 2 , for the discontinuous Galerkin finite element method for one-dimensional parabolic problems. Optimal convergence rates in both time and spatial variables are obtained. A discussion of automatic time-step control method is also included.

The space-time finite element method for parabolic problems

Adaptive space-time finite element method, continuous in space but discontinuous in time for semi-linear parabolic problems is discussed. The approach is based on a combination of finite element and finite difference techniques. The existence and uniqueness of the weak solution are proved without any assumptions on choice of the space-time meshes. Basic error estimates in L∞ (L2) norm, that is maximum-norm in time, L2-norm in space are obtained. The numerical results are given in the last part and the analysis between theoretic and experimental results are obtained.

Superconvergence of finite element method for parabolic problem

We study superconvergence of a semi‐discrete finite element scheme for parabolic problem. Our new scheme is based on introducing different approximation of initial condition. First, we give a superconvergence of uh − Rhu, then use a postprocessing to improve the accuracy to higher order.

Superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data

A semidiscrete mixed finite element approximation to parabolic initial-boundary value problems is introduced and analyzed. Superconvergence estimates for both pressure and velocity are obtained. The estimates for the errors in pressure and velocity depend on the smoothness of the initial data including the limiting cases of data in \(L^2\) and data in \(H^r\), for \(r\) sufficiently large. Because of the smoothing properties of the parabolic operator, these estimates for large time levels essentially coincide with the estimates obtained earlier for smooth solutions. However, for small time intervals we obtain the correct convergence orders for nonsmooth data.

The $p$ version of mixed finite element methods for parabolic problems

Un probleme parabolique a ete etudie d'un point de vue stabilite, ainsi que des proprietes d'approximation des methodes des elements finis mixtes (en espace) d'ordre croissant. Il est montre que des estimations precedentes faites sur la projection Raviart-Thomas sont a precises. Nous avons pu analyser les effets de la discretization du methode des elements finis, en espace, afin de presenter des estimations d'erreurs transitoires (pour des methodes des elements finis mixtes semidiscrets). Les resultats de cet article (soumis en Oct.1993) completent les resultats deja publies dans les references [6-8].

A semi-analytic boundary element method for parabolic problems

A new semi-analytic solution method is proposed for solving linear parabolic problems using the boundary element method. This method constructs a solution as an eigenfunction expansion using separation of variables. The eigenfunctions are determined using the dual reciprocity boundary element method. This separation of variables-dual reciprocity method (SOV-DRM) allows a solution to be determined without requiring either time-stepping or domain discretisation. The accuracy and computational efficiency of the SOV-DRM is found to improve as time increases. These properties make the SOV-DRM an attractive technique for solving parabolic problems.

Maximum Norm Analysis of Completely Discrete Finite Element Methods for Parabolic Problems

We consider full discretizations of linear parabolic initial value problems in any space dimension. Finite elements are used for the discretization in space. The resulting semidiscrete problem is integrated in time by means of a strongly $A(\theta )$-acceptable rational method. Stability estimates are derived and error bounds are established in the maximum norm for both smooth and rough initial data.

Adaptive Finite Element Methods for Parabolic Problems V: Long-Time Integration

We continue our previous work on adaptive finite element methods for parabolic problems, now with particular emphasis on long-time integration for semidefinite problems.

Adaptive Finite Element Methods for Parabolic Problems IV: Nonlinear Problems

We extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems. We prove a posteriori error estimates, design corresponding adaptive algorithms, and present some numerical results.

Adaptive Finite Element Methods for Parabolic Problems I: A Linear Model Problem

This paper is the first part in a series of papers on adaptive finite element methods for parabolic problems. In this paper, an adaptive algorithm is presented and analyzed for choosing the space and time discretization in a finite element method for a linear parabolic problem. The finite element method uses aspace discretization with meshsize variable in space and time and a third-order accurate time discretization with timesteps variable in time. The algorithm is proven to be (i) reliable in the sense that the $L_2 $-error in space is guaranteed to be below a given tolerance for all timesteps and (ii) efficient in the sense that the approximation error is for most timesteps not essentially below the given tolerance. The adaptive algorithm is based on an a posteriors error estimate which proves (i), and sharp a priori error estimates are used to prove (ii). Analogous results are given for the corresponding stationary (elliptic) problem. In the following papers in this series extensions are made, e.g., to timesteps variable also in space and to nonlinear problems.

Galerkin Finite Element Methods for Parabolic Problems.

The Standard Galerkin Method.- Methods Based on More General Approximations of the Elliptic Problem.- Nonsmooth Data Error Estimates.- More General Parabolic Equations.- Negative Norm Estimates and Superconvergence.- Maximum-Norm Estimates and Analytic Semigroups.- Single Step Fully Discrete Schemes for the Homogeneous Equation.- Single Step Fully Discrete Schemes for the Inhomogeneous Equation.- Single Step Methods and Rational Approximations of Semigroups.- Multistep Backward Difference Methods.- Incomplete Iterative Solution of the Algebraic Systems at the Time Levels.- The Discontinuous Galerkin Time Stepping Method.- A Nonlinear Problem.- Semilinear Parabolic Equations.- The Method of Lumped Masses.- The H1 and H?1 Methods.- A Mixed Method.- A Singular Problem.- Problems in Polygonal Domains.- Time Discretization by Laplace Transformation and Quadrature.

Galerkin finite element methods for parabolic problems, in lecture notes in mathematics, Vidar Thomée, Springer, Berlin, 1984

Galerkin finite element methods for parabolic problems, in <i>lecture notes in mathematics</i>, Vidar Thomée, Springer, Berlin, 1984

Negative norm estimates and superconvergence in Galerkin methods for parabolic problems

Negative norm error estimates for semidiscrete Galerkin-finite element methods for parabolic problems are derived from known such estimates for elliptic problems and applied to prove superconvergence of certain procedures for evaluating point values of the exact solution and its derivatives.