We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin (IPDG) methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l(H) and l(L) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin (SIPG) scheme with the implicit θ-schemes in time...

In this paper, an alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term. Firstly, one order implicit-explicit method is used for time discretization, then Galerkin finite element method is adopted for spatial discretization and obtain a fully discrete linear system. Secondly, Galerkin alternati...

In this paper, a fully discrete stabilized discontinuous Galerkin method is proposed to solve the Biot's consolidation problem. The existence and uniqueness of the finite element solution are obtained. The stability of the fully discrete solution is discussed. The corresponding error estimates for the approximation of displacement and pressure in a mesh dependent norm are obtained. The error es...

We reformulate the Johnson–Nedelec approach for the exterior two-dimensional Stokes problem taking advantage of the parameterization of the artificial boundary. The main aim of this paper is the presentation and analysis of a fully discrete numerical method for this problem. This one responds to the needs of having efficient approximate quadratures for the weakly singular boundary integrals. We...

2 Discretization 9 2.1 The Discontinuous Galerkin Method . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Solution Approximation . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.4 Discrete System . . . . . . . . . . . . . . . . . ....

2 Discretization 9 2.1 The Discontinuous Galerkin Method . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Solution Approximation . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.4 Discrete System . . . . . . . . . . . . . . . . . ....

In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play important roles in numerical methods for partial differential equations. This article intends to ...