DISCRETE ENERGY ANALYSIS OF THE THIRD-ORDER VARIABLE-STEP BDF TIME-STEPPING FOR DIFFUSION EQUATIONS

This is one of our series works on discrete energy analysis of the variable-step BDF schemes.In this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM J.Numer.Anal.,58:2294-2314]and[Math.Comp.,90:1207-1226]for our previous works on the BDF2 scheme.To this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation law.Mesh-robust stability and convergence analysis in the L^(2) norm are then obtained.Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios.We also present numerical tests to support our theoretical results.

ROSEN'S GRADIENT PROJECTION WITH DISCRETE STEPS

In 1960, J. B. Rosen gave a famous Gradient Projection Method in [1]. But the convergenceof the algorithm has not been proved for a Jong time. Many authors paid much attention to thisproblem, such as X.S. Zhang proved in [2] (1984) that the limit point of {x_k} which is generatedby Rosen's algorithm is a K-T piont for a 3-dimensional caes, if {x_k} is convergent. D. Z. Duproved in [3] (1986) that Rosen's algorithm is convergent for 4-dimensional.In [4] (1986), theauthor of this paper gave a general proof of the convergence of Rosen's Gradient Projection Methodfor an n-dimensional case. As Rosen's method requires exact line search, we know that exact linesearch is very difficult on computer.In this paper a line search method of discrete steps are presentedand the convergence of the algorithm is proved.

A Conservative Gradient Discretization Method for Parabolic Equations

In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and conservative gradient discretization method for spatial discretization.The method is based on a new cellcentered meshes,and it is locally conservative.It has smaller truncation error than the classical finite volume method on uniform meshes.We use the framework of the gradient discretization method to analyze the stability and convergence.The numerical experiments show that the new method has second-order convergence.Moreover,it is more accurate than the classical finite volume method in flux error,L2 error and L¥error.

A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation

A weak Galerkin(WG)method is introduced and numerically tested for the Helmholtz equation.This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property.At the same time,the WG finite element formulation is symmetric and parameter free.Several test scenarios are designed for a numerical investigation on the accuracy,convergence,and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains.Challenging problems with high wave numbers are also examined.Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement,and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.

A DISSIPATION-PRESERVING INTEGRATOR FOR DAMPED OSCILLATORY HAMILTONIAN SYSTEMS

In this paper,based on discrete gradient,a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established.The solution of this system is a damped nonlinear oscillator.Basically,lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach.The new integrator gives a discrete analogue of the dissipation property of the original system.Meanwhile,since the integrator is based on the variation-of-constants formula for oscillatory systems,it preserves the oscillatory structure of the system.Some properties of the new integrator are derived.The convergence is analyzed for the implicit iterations based on the discrete gradient integrator,and it turns out that the convergence of the implicit iterations based on the new integrator is independent of k Mk,where M governs the main oscillation of the system and usually k Mk≫1.This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system.Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature。

A Robust Modified Weak Galerkin Finite Element Method for Reaction-Diffusion Equations

In this paper,a robust modified weak Galerkin(MWG)finite element method for reaction-diffusion equations is proposed and investigated.An advantage of this method is that it can deal with the singularly perturbed reaction-diffusion equations.Another advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries freedom.It is worth pointing out that,in our method,the test functions space is the same as the finite element space,which is helpful for the error analysis.Optimalorder error estimates are established for the corresponding numerical approximation in various norms.Some numerical results are reported to confirm the theory.

A MODIFIED WEAK GALERKIN FINITE ELEMENT METHOD FOR SOBOLEV EQUATION

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete H^1 and L^2 norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.

ON L2 ERROR ESTIMATE FOR WEAK GALERKIN FINITE ELEMENT METHODS FOR PARABOLIC PROBLEMS

A weak Galerkin finite element method with stabilization term, which is symmetric, positive definite and parameter free, was proposed to solve parabolic equations by using weakly defined gradient operators over discontinuous functions. In this paper, we derive the optimal order error estimate in L2 norm based on dual argument. Numerical experiment is conducted to confirm the theoretical results.

TheWeak Galerkin Finite Element Method for Solving the Time-Dependent Integro-Differential Equations

In this paper,we solve linear parabolic integral differential equations using the weak Galerkin finite element method(WG)by adding a stabilizer.The semidiscrete and fully-discrete weak Galerkin finite element schemes are constructed.Optimal convergent orders of the solution of the WG in L^(2) and H^(1) norm are derived.Several computational results confirm the correctness and efficiency of the method.