Weak Galerkin finite element methods for Parabolic equations

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal‐order error estimates in both H¹ and L² norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Weak Galerkin finite element method for a class of time fractional generalized Burgers' equation

In this article, we use the weak Galerkin (WG) finite element method to study a class of time fractional generalized Burgers' equation. The existence of numerical solutions and the stability of fully discrete scheme are proved. Meanwhile, by applying the energy method, an optimal order error estimate in discrete L² norm is established. Numerical experiments are presented to validate the theoretical analysis.     

Weak Galerkin finite element method for a class of quasilinear elliptic problems

The purpose of this paper is to study the weak Galerkin finite element method for a class of quasilinear elliptic problems. The weak Galerkin finite element scheme is proved to have a unique solution with the assumption that guarantees the corresponding operator to be strongly monotone and Lipschitz-continuous. An optimal error estimate in a mesh-dependent energy norm is established. Some numerical results are presented to confirm the theoretical analysis.     

Weak Galerkin finite element methods for a fourth order parabolic equation

This paper is devoted to a newly developed weak Galerkin finite element method with the stabilization term for a linear fourth order parabolic equation, where weakly defined Laplacian operator over discontinuous functions is introduced. Priori estimates are developed and analyzed in L² and an H² type norm for both semi‐discrete and fully discrete schemes. And finally, numerical examples are provided to confirm the theoretical results.     

Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter‐free. Optimal order error estimates in a discrete H² norm is established for the corresponding WG finite element solutions. Error estimates in the usual L² norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014     

Some details of the Galerkin finite element method

The incorporation of the Galerkin technique in the finite element method has removed the constraint of finding a variational formulation for many problems of mathematical physics. The method has been successfully applied to many areas and has received wide acceptance. However, in the process of transplanting the concept from the Galerkin method for the entire domain to the Galerkin finite element method, some formal details have been overlooked or glossed over in the literature. This paper considers some of these details, including a possible reason for integration by parts and the contribution of interelement discontinuity terms.     

Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

In this paper, the weak Galerkin finite element method (WG-FEM) is applied to a pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. A fitted WG-FEM is proposed to approximate the voltage of the pulsed electric model across the physical media involving an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Optimal pointwise-in-time error estimates in L²-norm and H¹-norm are shown to hold for the semidiscrete scheme even if the regularity of the solution is low on the whole domain. Furthermore, a fully discrete approximation based on backward Euler scheme is analyzed and related optimal error estimates are derived.     

Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L²(H¹) and L²(L²) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition k_n≥ch², which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.     

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

Weak Galerkin finite element methods for linear parabolic integro‐differential equations

The semidiscrete and fully discrete weak Galerkin finite element schemes for the linear parabolic integro‐differential equations are proposed. Optimal order error estimates are established for the corresponding numerical approximations in both and norms. Numerical experiments illustrating the error behaviors are provided.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1357–1377, 2016     

Pricing of perpetual American and Bermudan options by binomial tree method

In this paper, we consider the binomial tree method for pricing perpetual American and perpetual Bermudan options. The closed form solutions of these discrete models are solved. Explicit formulas for the optimal exercise boundary of the perpetual American option is obtained. A nonlinear equation that is satisfied by the optimal exercise boundaries of the perpetual Bermudan option is found.      

Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form

A hybridization technique is applied to the weak Galerkin finite element method (WGFEM) for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results.     

Developing weak Galerkin finite element methods for the wave equation

In this article, we extend the recently developed weak Galerkin method to solve the second‐order hyperbolic wave equation. Many nice features of the weak Galerkin method have been demonstrated for elliptic, parabolic, and a few other model problems. This is the initial exploration of the weak Galerkin method for solving the wave equation. Here we successfully developed and established the stability and convergence analysis for the weak Galerkin method for solving the wave equation. Numerical experiments further support the theoretical analysis. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 868–884, 2017     

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

This article establishes a discrete maximum principle (DMP) for the approximate solution of convection–diffusion–reaction problems obtained from the weak Galerkin (WG) finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the WG involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin (SWG) method has a reduced computational complexity over the usual WG, and indeed provides a discretization scheme different from the WG when the reaction terms are present. An application of the SWG on uniform rectangular partitions yields some 5- and 7-point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the DMP and the accuracy of the scheme, particularly the finite difference scheme.     

Front-fixing FEMs for the pricing of American options based on a PML technique

In this paper, efficient numerical methods are developed for the pricing of American options governed by the Black–Scholes equation. The front-fixing technique is first employed to transform the free boundary of optimal exercise prices to some a priori known temporal line for a one-dimensional parabolic problem via the change of variables. The perfectly matched layer (PML) technique is then applied to the pricing problem for the effective truncation of the semi-infinite domain. Finite element methods using the respective continuous and discontinuous Galerkin discretization are proposed for the resulting truncated PML problems related to the options and Greeks. The free boundary is determined by Newton’s method coupled with the discrete truncated PML problem. Stability and nonnegativeness are established for the approximate solution to the truncated PML problem. Under some weak assumptions on the PML medium parameters, it is also proved that the solution of the truncated PML problem converges to that of the unbounded Black–Scholes equation in the computational domain and decays exponentially in the perfectly matched layer. Numerical experiments are conducted to test the performance of the proposed methods and to compare them with some existing methods.     

Superconvergence estimates of finite element methods for American options

In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation postprocessing analysis technique, we present sharp L ²-, L ^∞-norm error estimates and an H ¹-norm superconvergence estimate for this finite element method. As a by-product, the global superconvergence result can be used to generate an efficient a posteriori error estimator. This work was supported in part by the National Natural Science Foundation of China (10471103 and 10771158), the National Basic Research Program (2007CB814906), Social Science Foundation of the Ministry of Education of China (Numerical Methods for Convertible Bonds, 06JA630047), Tianjin Natural Science Foundation (07JCY-BJC14300), and Tianjin University of Finance and Economics.     

On discontinuous Galerkin finite element method for singularly perturbed delay differential equations

A nonsymmetric discontinuous Galerkin FEM with interior penalties has been applied to one-dimensional singularly perturbed problem with a constant negative shift. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Numerical experiments support theoretical findings.     

A C0‐weak Galerkin finite element method for fourth‐order elliptic problems

This article proposes and analyzes a C⁰‐weak Galerkin (WG) finite element method for solving the biharmonic equation in two‐dimensional and three‐dimensional. The new WG method uses continuous piecewise‐polynomial approximations of degree for the unknown u and discontinuous piecewise‐polynomial approximations of degree k for the trace of on the interelement boundaries. Optimal error estimates are obtained in H², H¹, and L² norms. Numerical experiments illustrate and confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1090–1104, 2016     

求解Brinkman方程的修正弱有限元方法

本文针对Brinkman方程引入了一种修正弱Galerkin(MWG)有限元方法.我们通过具有两个离散弱梯度算子的变分形式来逼近模型. 在MWG方法中, 分别用次数为$k$和$k-1$的不连续分段多项式来近似速度函数$u$和压力函数$p$. MWG方法的主要思想是用内部函数的平均值代替边界函数. 因此, 与WG方法相比, MWG方法在不降低准确性的同时, 具有更少的自由度, 对于任意次数不超过$k-1$ 的多项式,MWG方法均可以满足稳定性条件. MWG 方法具有高度的灵活性, 它允许在具有一定形状正则性的任意多边形或多面体上使用不连续函数. 针对$H^1$和$L^22$范数下的速度和压力近似解, 建立了最优阶误差估计. 数值算例表明了该方法的准确性, 收敛性和稳定性.