一阶非定常双曲问题的间断-差分流线扩散法及其误差估计

本文提出了求解一阶非定常双曲问题的一种新型有限元方法．间断-差分流线扩散法(DFDSD方法),建立了Euler型DFDSD格式,并对格式解的稳定性和收敛性进行了理论分析,最后给出了数值算例说明算法的有效性．     

A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation

Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal variable in both the element size h and polynomial degree p, and outperforms the standard upwind DG method.     

A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems

Abstract We derive a priori error estimates in the finite element method for nonselfadjoint elliptic and parabolic interface problems in a two-dimensional convex polygonal domain. Optimal H ¹-norm and sub-optimal L ²-norm error estimates are obtained for elliptic interface problems. For parabolic interface problems, the continuous-time Galerkin method is analyzed and an optimal order error estimate in the L ²(0,T;H ¹)-norm is established. Further, a discrete-in-time discontinuous Galerkin method is discussed and a related optimal error estimate is obtained. Keywords: Elliptic and parabolic interface problems, finite element method, spatially discrete scheme, discontinuous Galerkin method, error estimates Mathematics Subject Classification (1991): 65N15, 65N20     

Superconvergence of Discontinuous Finite Element Solutions for Transient Convection–diffusion Problems

We present a study of the local discontinuous Galerkin method for transient convection–diffusion problems in one dimension. We show that p-degree piecewise polynomial discontinuous finite element solutions of convection-dominated problems are O(Δx ^p+2) superconvergent at Radau points. For diffusion- dominated problems, the solution’s derivative is O(Δx ^p+2) superconvergent at the roots of the derivative of Radau polynomial of degree p+1. Using these results, we construct several asymptotically exact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically exact.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Characteristic Tailored Finite Point Method for Convection-Dominated Convection-Diffusion-Reaction Problems

In this paper, we propose a characteristic tailored finite point method (CTFPM) for solving the convection-diffusion-reaction equation with variable coefficients. We develop an algorithm to construct a streamline-aligned grid for the CTFPM. Our numerical tests show for small diffusion coefficient the CTFPM solution resolves the internal and boundary layers regardless the mesh size, and depicts that CTFPM method with a streamline grid has excellent performance compared with the tailored finite point method and a streamline upwind finite element method when ε is small.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

Convergence of an upwind control-volume mixed finite element method for convection–diffusion problems

Summary We consider a upwind control volume mixed finite element method for convection–diffusion problem on rectangular grids. These methods use the lowest order Raviart–Thomas mixed finite element space as the trial functional space and associate control-volumes, or covolumes, with the vector variable as well as the scalar variable. Chou et al. [6] established a one-half order convergence in discrete L ²-norms. In this paper, we establish a first order convergence for both the vector variable as well as the scalar variable in discrete L ²-norms.      

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(h^k+1+τ^2?α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.     

Finite Volume Element Approximations of Nonlocal in Time One-Dimensional Flows in Porous Media

Various finite volume element schemes for parabolic integro-differential equations in 1-D are derived and studied. These types of equations arise in modeling reactive flows or material with memory effects. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. We consider the lowest-order (linear and L-splines) finite volume elements, although higher-order volume elements can be considered as well under this framework. It is proved that finite volume element approximations are convergent with optimal order in H ¹-norms, suboptimal order in the L ²-norm and super-convergent order in a discrete H ¹-norm. Received August 3, 1998; revised October 11, 1999     

Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.     

Quasi-Compact Finite Difference Schemes for Space Fractional Diffusion Equations

In this paper, a compact difference operator, termed CWSGD, is designed to establish the quasi-compact finite difference schemes for approximating the space fractional diffusion equations in one and two dimensions. The method improves the spatial accuracy order of the weighted and shifted Grünwald difference (WSGD) scheme (Tian et al., arXiv:1201.5949) from 2 to 3. The numerical stability and convergence with respect to the discrete L ² norm are theoretically analyzed. Numerical examples illustrate the effectiveness of the quasi-compact schemes and confirm the theoretical estimations.     

Nodal Superconvergence of SDFEM for Singularly Perturbed Problems

In this paper, we analyze the streamline diffusion finite element method for one dimensional singularly perturbed convection-diffusion-reaction problems. Local error estimates on a subdomain where the solution is smooth are established. We prove that for a special group of exact solutions, the nodal error converges at a superconvergence rate of order (ln ε ⁻¹/N)^2k (or (ln N/N)^2k ) on a Shishkin mesh. Here ε is the singular perturbation parameter and 2N denotes the number of mesh elements. Numerical results illustrating the sharpness of our theoretical findings are displayed.     

Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem

In this paper, a discontinuous finite volume element method was presented to solve the nonstationary Stokes–Darcy problem for the coupling fluid flow in conduits with porous media flow. The proposed numerical method is constructed on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by piecewise constant elements. The unique solvability of the approximate solution for the discrete problem is derived. Optimal error estimates of the semi-discretization and full discretization with backward Euler scheme in standard \(L^2\)-norm and broken \(H^1\)-norm are obtained for three discontinuous finite volume element methods (symmetric, non-symmetric and incomplete types). A series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, mass conservation, capability to deal with complicated geometries, and applicability to the problems with realistic parameters.     

Local Discontinuous Galerkin Method for Surface Diffusion and Willmore Flow of Graphs

In this paper, we develop a local discontinuous Galerkin (LDG) finite element method for surface diffusion and Willmore flow of graphs. We prove L ² stability for the equation of surface diffusion of graphs and energy stability for the equation of Willmore flow of graphs. We provide numerical simulation results for different types of solutions of these two types of the equations to illustrate the accuracy and capability of the LDG method.     

Guaranteed and Fully Robust a posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients

We study in this paper a posteriori error estimates for H ¹-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.     

On fully discrete schemes for the Fermi pencil-beam equation

We consider a Fermi pencil-beam model in two-space dimensions (x,y), where x is aligned with the beam’s penetration direction and y together with the scaled angular variable z correspond to a, bounded symmetric, transversal cross-section. The model corresponds to a forward–backward degenerate, convection dominated, convection–diffusion problem. For this problem we study some fully discrete numerical schemes using the standard- and Petrov–Galerkin finite element methods, for discretizations of the transversal domain, combined with the backward Euler, Crank–Nicolson, and discontinuous Galerkin methods for discretizations in the penetration variable. We derive stability estimates for the semi-discrete problems. Further, assuming sufficiently smooth exact solution, we obtain optimal a priori error bounds in a triple norm. These estimates give rise to a priori error estimates in the L²-norm. Numerical implementations presented for some examples with the data approximating Dirac δ function, confirm the expected performance of the combined schemes.     

A transient finite element solution method for the Navier-Stokes equations

This paper is concerned with the discrete finite element formulation and numerical solution of transient incompressible viscous flow in terms of the primitive variables. A restricted variational principle is introduced as equivalent to the momentum equations and the Poisson equation for pressure. The latter is introduced to replace the continuity equation, and thus the incompressibility condition is realized only asymptotically; i.e. through the iterative process. An incomplete cubic interpolation function is used for both the velocities and pressure within a triangular finite element. The discrete equations are integrated in time with backward finite differences. We illustrate the similarity between the (ψ,ζ) finite difference method and the ($\text{u}\text{,p}$) finite element method by calculations on the driven square cavity problem.     

Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation

We present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O( H ³/h ³) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincaré–Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces.This revised version was published online in July 2005 with corrected volume and issue numbers.     

On the Stability of Continuous–Discontinuous Galerkin Methods for Advection–Diffusion–Reaction Problems

We consider a finite element method which couples the continuous Galerkin method away from internal and boundary layers with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. The stability properties of the coupled method are illustrated with a numerical experiment.     

An assessment of discretizations for convection-dominated convection–diffusion equations

The performance of several numerical schemes for discretizing convection-dominated convection–diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov–Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.