Relaxed one-sided Lipschitz conditions play an important role when analyzing ordinary differential inclusions. They allow to derive a-priori estimates of solutions and convergence estimates for explicit and implicit time discretizations. In this paper we consider Galerkin finite element discretizations of semilinear elliptic inclusions that satisfy a relaxed one-sided Lipschitz condition. It is...

In this pat)er we introduce a high order discontinuous Galerkin method for two dimensional incoinpressible flow in vorticity streamfunction fornnllation. The inonlentuni equation is treated exl)licitly, utilizing the efficiency of the discontimtous Galerkin method. The streanlflmction is obtained by a standard Poiss(m solver using (:ontinu(lus finite elenmnts. There is a natural matching betwee...

This paper discusses the numerical solution of Fredholm integral equations of the second kind which have weakly singular kernels and inhomogeneous terms. Global convergence estimates are derived for the Galerkin and iterated Galerkin methods using splines on arbitrary quasiuniform meshes as approximating subspaces. It is observed that, due to the singularities present in the solution being appr...

A space-time Galerkin/least-squares finite element method was presented in [l] for numerical simulation of single-carrier hydrodynamic semiconductor device equations. The single-carrier hydrodynamic device equations were shown to resemble the ideal gas equations and Galerkin/least-squares finite element method, originally developed for computational fluid dynamics equations [16], was extended t...

DarcyLite is a Matlab toolbox for numerical simulations of flow and transport in 2-dim porous media. This paper focuses on the finite element (FE) methods and the corresponding code modules in DarcyLite for solving the Darcy equation. Specifically, four major types of finite element solvers, the continuous Galerkin (CG), the discontinuous Galerkin (DG), the weak Galerkin (WG), and the mixed fin...

In this paper, we uncover and study a new superconvergence property of a large class of finite element methods for one-dimensional convectiondiffusion problems. This class includes discontinuous Galerkin methods defined in terms of numerical traces, discontinuous Petrov–Galerkin methods and hybridized mixed methods. We prove that the so-called numerical traces of both variables superconverge at...

In this paper, we investigate a new approach for the numerical solution of the two-dimensional depth-integrated shallow water equations, based on coupling discontinuous and continuous Galerkin methods. In this approach, we couple a discontinuous Galerkin method applied to the primitive continuity equation, coupled to a continuous Galerkin method applied to the so-called ‘‘wave continuity equati...

Abstract. It is proven that a class of Gegenbauer tau approximations to a 4th order differential eigenvalue problem of hydrodynamic type provide real, negative, and distinct eigenvalues as for the exact solutions. This class of Gegenbauer tau methods includes Chebyshev and Legendre Galerkin and ‘inviscid’ Galerkin but does not include Chebyshev and Legendre tau. Rigorous and numerical results s...

A three-step wavelet Galerkin method based on Taylor series expansion in time is proposed. The scheme is third-order accurate in time and O(2−jp) accurate in space. Unlike Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving non-linear problems. The compactly supported orthogonal wavelet bases D6 developed by Daubechie...

The discretization of linear partial differential equations with random data by means of the stochastic Galerkin finite element method results in general in a large coupled linear of system of equations. Using the stochastic diffusion equation as a model problem, we introduce and study a symmetric positive definite Kronecker product preconditioner for the Galerkin matrix. We compare the popular...