A MODIFIED WEAK GALERKIN FINITE ELEMENTMETHOD FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION-REACTION PROBLEMS

In this work,a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations.The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators,respectively.We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh.The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods.Optimal order of convergences are obtained in suitable norms.We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method.Various numerical examples are presented to support the theoretical results.It is theoretically and numerically shown that the method is quite stable.

A hybridized weak Galerkin finite element scheme for the Stokes equations

In this paper a hybridized weak Galerkin(HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is introduced.The WG method uses weak functions and their weak derivatives which are defined as distributions.Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees.Different combination of polynomial spaces leads to different WG finite element methods,which makes WG methods highly flexible and efficient in practical computation.A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact solution.With this new feature,the HWG method can be used to deal with jumps of the functions and their flux easily.Optimal order error estimates are established for the corresponding HWG finite element approximations for both primal variables and the Lagrange multiplier.A Schur complement formulation of the HWG method is derived for implementation purpose.The validity of the theoretical results is demonstrated in numerical tests.

Weak Galerkin finite element method for valuation of American options

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.

A WEAK GALERKIN FINITE ELEMENT METHOD FOR THE LINEAR ELASTICITY PROBLEM IN MIXED FORM

In this paper, we use the weak Galerkin （WG） finite element method to solve the mixed form linear elasticity problem. In the mixed form, we get the discrete of proximation of the stress tensor and the displacement field. For the WG methods, we define the weak function and the weak differential operator in an optimal polynomial approximation spaces. The optimal error estimates are given and numerical results are presented to demonstrate the efficiency and the accuracy of the weak Galerkin finite element method.

AStabilizer-FreeWeak Galerkin Finite Element Method for the Stokes Equations

A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper.Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces for the weak gradient operators.The new algorithm is simple in formulation and the computational complexity is also reduced.The corresponding approximating spaces consist of piecewise polynomials of degree k≥1 for the velocity and k-1 for the pressure,respectively.Optimal order error estimates have been derived for the velocity in both H^(1) and L^(2) norms and for the pressure in L^(2) norm.Numerical examples are presented to illustrate the accuracy and convergency of the method.

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.

A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

In this article,a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed.The idea of using the P_(k)-harmonic polynomial space instead of the full polynomial space P_(k)is to use a much smaller number of basis functions to achieve the same accuracy when k≥2.The optimal rate of convergence is derived in both H^(1)and L^(2)norms.Numerical experiments have been conducted to verify the theoretical error estimates.In addition,numerical comparisons of using the P_(2)-harmonic polynomial space and using the standard P_(2)polynomial space are presented.

A weak Galerkin-mixed finite element method for the Stokes-Darcy problem

In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation.A discrete inf-sup condition is proved and the optimal error estimates are also derived.Numerical experiments validate the theoretical analysis.

The Weak Galerkin Method for Elliptic Eigenvalue Problems

This article is devoted to studying the application of the weak Galerkin(WG)finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds.The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions.The non-conforming finite element space of the WG method is the key of the lower bound property.It also makes the WG method more robust and flexible in solving eigenvalue problems.We demonstrate that the WG method can achieve arbitrary high convergence order.This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements.Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.

The Weak Galerkin Method for Linear Hyperbolic Equation

The linear hyperbolic equation is of great interest inmany branches of physics and industry.In this paper,we use theweak Galerkinmethod to solve the linear hyperbolic equation.Since the weak Galerkin finite element space consists of discontinuous polynomials,the discontinuous feature of the equation can be maintained.The optimal error estimates are proved.Some numerical experiments are provided to verify the efficiency of the method.

THE SHIFTED-INVERSE POWER WEAK GALERKIN METHOD FOR EIGENVALUE PROBLEMS

This paper proposes and analyzes a new weak Galerkin method for the eigenvalue problem by using the shifted-inverse power technique.A high order lower bound can be obtained at a relatively low cost via the proposed method.The error estimates for both eigenvalue and eigenfunction are provided and asymptotic lower bounds are shown as well under some conditions.Numerical examples are presented to validate the theoretical analysis.

On the superconvergence of a WG method for the elliptic problem with variable coefficients

This article extends a recently developed superconvergence result for weak Galerkin(WG)approximations for modeling partial differential equations from constant coefficients to variable coefficients.This superconvergence features a rate that is two orders higher than the optimal-order error estimates in the usual energy and L^(2)norms.The extension from constant to variable coefficients for the modeling equations is highly non-trivial.The underlying technical analysis is based on a sequence of projections and decompositions.Numerical results confirm the superconvergence theory for second-order elliptic problems with variable coefficients.

A Numerical Analysis of the Weak Galerk in Method for the Helmholtz Equation with High Wave Number

We study the error analysis of the weak Galerkin finite element method in[24,38](WG-FEM)for the Helmholtz problem with large wave number in two and three dimensions.Using a modified duality argument proposed by Zhu and Wu,we obtain the pre-asymptotic error estimates of the WG-FEM.In particular,the error estimates with explicit dependence on the wave number k are derived.This shows that the pollution error in the broken H1-norm is bounded by O(k(kh)^(2p))under mesh condition k^(7/2)h^(2)≤C0 or(kh)^(2)+k(kh)^(p+1)≤C_(0),which coincides with the phase error of the finite element method obtained by existent dispersion analyses.Here h is the mesh size,p is the order of the approximation space and C_(0) is a constant independent of k and h.Furthermore,numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the WG-FEM in reducing the pollution effect.