Method For Parabolic Problems Research Articles

A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems

We consider fully discrete time-space approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an hp-version discontinuous Galerkin (DG) time stepping scheme in conjunction with standard (conforming) Galerkin discretizations in space. We derive abstract computable a posteriori error bounds resulting, for instance, in concrete bounds in - and -type norms when I is the temporal and the spatial domain for the PDE. We base our methodology for the analysis on a novel space-time reconstruction approach. Our approach is flexible as it works for any type of elliptic error estimator and leaves their choice to the user. It also exhibits mesh-change estimators in a clear and concise way. We also show how our approach allows the derivation of such bounds in the norm.

A Multiscale Method for Heterogeneous Bulk-Surface Coupling

In this paper, we construct and analyze a multiscale (finite element) method for parabolic problems with heterogeneous dynamic boundary conditions. As the origin, we consider a reformulation of the system in order to decouple the discretization of bulk and surface dynamics. This allows us to combine multiscale methods on the boundary with standard Lagrangian schemes in the interior. We prove convergence and quantify explicit rates for low-regularity solutions, which are independent of the oscillatory behavior of the heterogeneities. As a result, coarse discretization parameters, which do not resolve the fine scales, can be considered. The theoretical findings are justified by a number of numerical experiments including dynamic boundary conditions with random diffusion coefficients.

Convergence in Wavelet Collocation Methods for Parabolic Problems

Convergence in Wavelet Collocation Methods for Parabolic Problems

Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data

Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data

A FETI-based mixed explicit–implicit multi-time-step method for parabolic problems

Nonstationary partial differential equations are numerically solved by discretizing in space and then integrating over time using discrete solvers. In this paper we propose and examine a mixed subcycling time stepping strategy using FETI domain decomposition for parabolic problems (e.g. transient heat conduction). The computational domain is divided into a set of smaller subdomains that may be integrated sequentially with its own time steps and generalized trapezoidal α -methods. The continuity condition at the interface is ensured using a dual Schur complement formulation. The rigorous stability analysis of the proposed algorithm is performed via the energy method. It was proved that the method is unconditionally stable provided α k ≥ 1 ∕ 2 in all subdomains Ω k . Moreover, the same analysis indicates that the mixed explicit/implicit Euler method is conditionally stable. Some example problems are presented to examine the rate of convergence, stability as well as accuracy of the mixed multi-time step algorithm.

A new approach to constructing vector splitting schemes in mixed finite element method for parabolic problems

Abstract A general setting for a new approach to constructing vector splitting schemes in mixed FEM for heat transfer problem is considered. For space approximation Raviart–Thomas finite elements of lowest order are implemented on rectangular (parallelepiped) mesh. The main question discussed is how to implement time discretization so as to obtain efficient numerical algorithms. The key idea of the proposed approach is to use scalar splitting schemes for heat flux divergence. This allows one to carry out accuracy and stability analysis on the basis of the well-known results for underlying scalar splitting schemes. A priori estimates are obtained for vector splitting schemes in two- and three-dimensional cases using the idea of flux decomposition onto divergence-free and potential components at discrete level. It is shown that splitting schemes which have been proposed by other approaches based on Uzawa algorithm’s modification and block triangular factorization are included as special cases by an appropriate choice of the underlying splitting schemes for heat flux divergence. Generalization of the discussed approach is presented for second-order hyperbolic problems. Moreover, the general idea of using splitting schemes for the flux divergence to construct splitting schemes for fluxes can be easily extended to first-order hyperbolic systems, but by now several questions still remain unsolved.

Space–time adaptive linearly implicit peer methods for parabolic problems

In this paper a linearly implicit peer method is combined with a multilevel finite element method for the discretization of parabolic partial differential equations. Following the Rothe method it is first discretized in time and then in space. A spatial error estimator based on the hierarchical basis approach is derived. It is shown to be a reliable and efficient estimator up to some small perturbations. The efficiency index of the estimator is shown to be close to the ideal value one for two one-dimensional test problems. Finally we compare the performance of the overall method, based on second, third, and fourth order peer methods with that of some Rosenbrock methods. We conclude that the presented peer methods offer an attractive alternative to Rosenbrock methods in this context.

Convergence of H 1-Galerkin mixed finite element method for parabolic problems with reduced regularity on initial data

We study the convergence of H1-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming less regularity on initial data. More precisely, for the spatially discrete scheme, error estimates of order \(\mathcal{O}\)(h2t−1/2) for positive time are established assuming the initial function p0 ∈ H2(Ω) ∩ H01 (Ω). Further, we use energy technique together with parabolic duality argument to derive error estimates of order \(\mathcal{O}\)(h2t−1) when p0 is only in H01 (Ω). A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established.

Convergence of the Swartztrauber-Sweet Method for Parabolic Problems on a Disk with Singularities

In this article, we study the convergence analysis for the initial and boundary value problem of parabolic equations on a disk with singular solutions. It is assumed that the exact solution performs singular properties that its derivatives go to infinity at the boundary of the disk. We propose a fully implicit time-stepping numerical scheme. A stretching polynomial-like function with a parameter is used to construct a local grid refinement. Over the nonuniform partition, we combine the Swartztrauber-Sweet scheme and the backward Euler method in spatial and temporal discretization, respectively. We carry out convergence analysis and analyze the effects of the parameter. It is shown that our numerical scheme is of first order accuracy for temporal discretization and of almost second order accuracy for spatial discretization. Numerical experiments are performed to illustrate our analysis results and show that there exists an optimal value for the parameter to obtain a best approximate solution.

Discontinuous mixed covolume methods for parabolic problems.

We present the semidiscrete and the backward Euler fully discrete discontinuous mixed covolume schemes for parabolic problems on triangular meshes. We give the error analysis of the discontinuous mixed covolume schemes and obtain optimal order error estimates in discontinuous H(div) and first-order error estimate in L 2.

A Fully Discrete Stabilized Mixed Finite Element Method for Parabolic Problems

In this article, we present a new stabilized mixed finite element method based on the less regularity of flux (velocity) for the two-dimensional parabolic problems in practice. The method combines the Crank-Nicolson scheme with a stabilized mixed finite element method which is based on the velocity projection stabilization method by using the lowest equal-order pair for the velocity and pressure. It is shown that the proposed fully discrete stabilized finite element method results in the optimal error estimates in L 2- and H 1-norm for the pressure and suboptimal error estimate in L 2-norm for the velocity. Finally, we give some numerical experiments to verify the efficiency and theoretical results of this method.

A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H 1-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.

Laplace Transform Method for Parabolic Problems with Time-Dependent Coefficients

The Laplace transform method has proven to be very efficient for dealing with parabolic problems whose coefficients are time independent, and it is easily parallelizable. However, the method has not been proven to be applicable to linear problems whose coefficients are time dependent. The reason is that the Laplace transform of two time-dependent functions leads to a convolution of the Laplace transformed functions in the dual variable. In this paper, we propose a Laplace transform method to linear parabolic problems with time-dependent coefficients, which is as efficient as the method for parabolic problems with time-independent coefficients. Several numerical results are provided, which support the efficiency of the proposed scheme.

Multiscale computation method for parabolic problems of composite materials

In this paper, we consider the initial-boundary value problem of parabolic type equation with rapidly oscillating coefficients in both time and space. A multiscale asymptotic expansion of solution for this kind of problem is presented. The full discrete finite element method for computing above problem is introduced. This method can apply to heat conduction analysis of composite materials. The main advantages of this method are that it can greatly save computer memory and CPU time, and it has good precision at the same time. Finally numerical results show that the method presented in this paper is effective and reliable.

Higher-order schemes for the Laplace transformation method for parabolic problems

In this paper we solve linear parabolic problems using the three stage noble algorithms. First, the time discretization is approximated using the Laplace transformation method, which is both parallel in time (and can be in space, too) and extremely high order convergent. Second, higher-order compact schemes of order four and six are used for the the spatial discretization. Finally, the discretized linear algebraic systems are solved using multigrid to show the actual convergence rate for numerical examples, which are compared to other numerical solution methods.

Second‐order treatment of the interface of domain decomposition method for parabolic problems

AbstractEstimations for the values on interface lines are necessary in a domain decomposition method. However, the accuracy of the estimations is of the first order for most of unconditionally stable domain decomposition schemes. In this paper, a second order of accuracy for the estimations on interface lines is presented. With the new scheme, the optimal number of decomposed subdomains has been observed and proposed. Moreover when the SOR iterative method is used, the optimal over‐relaxation parameter is also studied. Copyright © 2010 John Wiley & Sons, Ltd.

Acceleration of a Schwarz waveform relaxation method for parabolic problems

In this paper we generalise the Aitken-like acceleration method of the additive Schwarz algorithm for elliptic problems to the additive Schwarz waveform relaxation (ASWR) for parabolic problems. The domain decomposition is in space and time. We show that our technique: 1) is a direct solver that requires at most four solves per subdomain in the case of a one space dimension linear parabolic problem with time independent coefficients; 2) can be applied easily to multidimensional problems, provided that the operator is separable in space; 3) is an efficient iterative procedure for parabolic problems that are weak non-linear perturbations of linear operators with time independent coefficients; 4) provides a rigorous framework to optimise the parallel implementation on a slow network of computers.

A high-order ADI method for parabolic problems with variable coefficients

A high-order compact alternating direction implicit (ADI) method is proposed for solving two-dimensional (2D) parabolic problems with variable coefficients. The computational problem is reduced to sequence one-dimensional problems which makes the computation cost-effective. The method is easily extendable to multi-dimensional problems. Various numerical tests are performed to test its high-order accuracy and efficiency, and to compare it with the standard second-order Peaceman–Rachford ADI method. The method has been applied to obtain the numerical solutions of the lid-driven cavity flow problem governed by the 2D incompressible Navier–Stokes equations using the stream function-vorticity formulation. The solutions obtained agree well with other results in the literature.

Discontinuous finite volume element method for parabolic problems

Discontinuous finite volume element method for parabolic problems

Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations

A smoothing property in multistep backward difference method for a linear parabolic problem in Hilbert space has been proved, where the operator is selfadjoint, positive definite with compact inverse. By using the solutions computed by a multistep backward difference method for the parabolic problem, we introduce an approximation scheme for time derivative. The nonsmooth data error estimate for the approximation of time derivative has been obtained.