In the numerical simulation for nonlinear diffusion problems with degenerate diffusion coefficients, some classical methods are often invalid since they involve a harmonic mean of diffusion coefficients on the adjacent cells. To avoid such problem, we consider using linear finite element method to solve a class of 1D degenerate nonlinear parabolic equations. This method can effectively capture the profile of true solution, but at the front it generates some nonphysical numerical oscillations, even brings forth negative values in numerical solution for approximating nonnegative physical quantities. In order to preserve the nonnegativity of true solution, we discuss three repair techniques for finite element solutions based on a posteriori corrections. The first one is a zero-setting method, in which we directly set those negative values to be zero. The second one is a local approach, in which any negative energy associate to some node is absorbed by the positive values around the current node. The third one is a global strategy, in which the total negative energy is redistributed to all positive values with a one-time effort. Numerical examples show that the numerical solution profile is improved remarkably by using the repair techniques for degenerate parabolic equations.

The classical Black-Scholes pricing model is based on standard geometric Brownian motion, and the log-returns of this model are independent and Gaussian. However, most of the recent researches on the statistical properties of the log-returns make this hypothesis not always consistent. One of the ongoing issues of mathematical finance today is to design an efficient numerical algorithm for the pricing model, which might be modified from the standard Black-Scholes diffusion equation and would have favorable empirical results.
Of those financial models that have been already proposed, the most interesting include the Finite Moment Log-Stable (FMLS) process model and its fractional partial integral-differential equation. In this paper, we consider using Gauss-Jacobi spectral method on a two-dimensional computation domain in order to discretize the FMLS fractional partial integral-differential equation, and further illustrate the flexibility and accuracy of the method by comparing the first order finite difference scheme for the pricing examples of European and American-styled options. Our results suggest that the global character of the Gauss-Jacobi method makes them well-suited to fractional partial integral-differential equations and can naturally take the global behavior of the solution into account and thus do not lead to an extra computational cost when moving from a second-order to a fractional-order diffusion model.

In this paper we are concerned with numerical methods for nonhomogeneous Helmholtz equations in inhomogeneous media. We try to extend the plane wave method, which has been widely used to the discretization of homogeneous Helmholtz equations with constant wave numbers, to solve nonhomogeneous Helmholtz equations with piecewise constant wave numbers. To this end, we propose a combination between the plane wave discontinuous Galerkin (PWDG) method and the high order element discontinuous Galerkin (HODG) method for the underlying Helmholtz equations. In this composite methods, we need only to solve a series of local Helmholtz equations by the HODG method and solve a locally homogeneous Helmholtz equation with element-wise constant wave number by the PWDG method. In particular, we first consider non-matching grids so that the generated approximations have better performance for the current case. Numerical experiments show that the proposed methods are very effective to the tested Helmholtz equations.

This paper concerns dissipativity of one-leg θ-methods and linear θ-methods for nonlinear delay differential equations (DDEs). Firstly, we obtain the absorbing set generated by the numerical methods and then prove that the methods can inherit the dissipativity of underlying system. It is shown that the radius of the absorbing set generated by the discrete system goes to that generated by the continuous system as the step size goes to zero. The estimate of the radius is sharp in this sense. Secondly, because the model considered is a very broad class of differential equations with or without delays, the main results obtained provide a unified treatment for ordinary differential equations (ODEs) and DDEs with constant delays and variable delays (including bounded and unbounded variable delays). In particular, the results are also new even in the case of ODEs. Finally, numerical experiments are given to support our theoretical results.

An effective method is proposed for interface problems by combining the domain decomposition method and the collocation method. The main idea is to split the whole domain into two non-overlapped sub-domains and enforce the interface conditions to obtain two sub-problems. Each sub-problem is solved by the collocation method. And a simple iterative algorithm is presented to achieve the jump conditions at the interface. The method provided can be used to solve interface problems with both linear and a special type of non-linear jump conditions. Numerical experiments reveal that our method is fourth-order accurate for interface problems.

The objective of this paper is to enlarge the ball of convergence and improve the error bounds of the modified Newton method for solving equations with solutions of multiplicity under weak conditions.

A modified weak Galerkin finite element method is applied to the poroelasticity problems, in which, we use the piecewise polynomial space to approximate the displacement and the pressure, and we utilize the weak derivative operators to replace the classical ones in the modified weak Galerkin algorithm. Based on the traditional weak Galerkin finite element method, the modified method reduces the total amount of computation by eliminating the degrees of freedom on the boundaries. The error estimates are given and the numerical results are reported to illustrate our theoretical results.

In the current investigation, we present a numerical technique to solve Fredholm-Hammerstein integral equations of the second kind. The method utilizes the free shape parameter radial basis functions (RBFs) constructed on scattered points as a basis in the discrete Galerkin method to estimate the solution of integral equations. The accuracy and stability of the classical RBFs heavily depend on the selection of shape parameters. But on the other hand, the choice of suitable value for shape parameters is very difficult. Therefore, to get rid of this problem, the free shape parameter RBFs are used in the new method which establish an effective and stable method to estimate an unknown function. We utilize the composite Gauss-Legendre integration rule and employ it to estimate the integrals appeared in the method. Since the scheme does not need any background meshes, it can be identified as a meshless method. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.

Hexagonal grid methods are valuable in two-dimensional applications involving Laplacian. The methods are investigated on problems related to standard and anisotropic Laplacian using Fourier vectors in pure, mixed and combination types. Complete (positive) eigenvalues and eigenvectors are determined explicitly in terms of various bases in a unified structure. This work is the smallest completion of some previous works.

The weak Galerkin finite element method is a class of recently and rapidly developed numerical tools for approximating partial differential equations. Unlike the standard Galerkin method, its trial and test function spaces consist of totally discontinuous piecewisely defined polynomials in the whole domain. This method has been vastly applied to many fields [22, 28, 31, 44, 50-52]. In this paper, we will apply this method to approximate a stochastic parabolic partial differential equation. We set up a semi-discrete numerical scheme for the stochastic partial differential equations and derive the optimal order for error estimates in the sense of strong convergence.

In this article we are interested in the semi-group stability for finite difference schemes approximations of hyperbolic systems of equations in corner domains. We give generalizations of the results of [10] and [9] from the half space geometry to the quarter space geometry. The most interesting fact is that the proofs of [10] and [9] can be adaptated with minor changes to apply in the quarter space geometry. This is due to the fact that both methods in [10] and [9] are based on energy methods and the construction of auxiliary problems with strictly dissipative boundary conditions which are known to be suitable for the strong well-posed for initial boundary value problems in the quarter space.

This paper is concerned with numerical approximation of elliptic interface problems via week Galerkin (WG) finite element method. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. In the implementation, the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the corresponding WG scheme are significantly impacted by the selection of such polynomials. This paper presents an optimal combination for the polynomial spaces that minimizes the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both $H^1$ and $L^2$ norms are established for the present WG finite element solutions.