In this paper we introduce a new method for solving partial and ordinary differential equations with large first, second and third derivatives of the solution in some part of the domain using the finite element technique (here called the Galerkin-Gokhman method). The method is based on the application of the Galerkin method to a modified differential equation. The exact solution of the modified...

In this paper, we study the numerical approximation of the Urysohn integral equation with two methods. The methods are developed by means of the sinc approximation with the Single Exponential (SE) and Double Exponential (DE) transformations. These numerical methods combine a sinc Nyström method with the Newton iterative process that involves solving a nonlinear system of equations. We provide a...

In this paper we give stability analysis and error estimates for the recently introduced central discontinuous Galerkin method when applied to linear hyperbolic equations. A comparison between the central discontinuous Galerkin method and the regular discontinuous Galerkin method in this context is also made. Numerical experiments are provided to validate the quantitative conclusions from the a...

Proliferation of degrees-of-freedom has plagued discontinuous Galerkin methodology from its inception over 30 years ago. This paper develops a new computational formulation that combines the advantages of discontinuous Galerkin methods with the data structure of their continuous Galerkin counterparts. The new method uses local, element-wise problems to project a continuous finite element space ...

We study a multiscale discontinuous Galerkin method introduced in [10] that reduces the computational complexity of the discontinuous Galerkin method, seemingly without adversely affecting the quality of results. For a stabilized variant we are able to obtain the same error estimates for the advection-diffusion equation as for the usual discontinuous Galerkin method. We assess the stability of ...