A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations

A PRIORI L2ERROR ANALYSIS FOR AN EXPANDED MIXED FINITE ELEMENT METHOD FOR QUASILINEAR PSEUDO-PARABOLIC EQUATIONS

“…By applying (3.3), (3.1) and (2.5) to (3.15) we have 22) which proves that (τ , ρ) satisfies (3.9). Therefore we conclude that the fixed point (τ , η, ρ) ∈ V h × Λ h × W h of Φ is the solution of (3.7)-(3.9).…”

“…Early many authors applied classical Galerkin methods [1,2,16,17,18] or discontinuous Galerkin methods [22,23] to construct the semidiscrete or fully discrete approximations to the solutions of the various types of the pseudoparabolic equations. Compared to the classical Galerkin finite element method, the advantage of mixed finite element methods (MFEM) is to compute simultaneously the finite element approximations of the flux and the unknown scalar without requiring the additional regularities of u(x, t).…”

L2-ERROR ANALYSIS OF FULLY DISCRETE DISCONTINUOUS GALERKIN APPROXIMATIONS FOR NONLINEAR SOBOLEV EQUATIONS

“…Lin and Zhang [14] constructed the semidiscrete finite element approximations and derived the optimal L 2 error estimates with nonlinear boundary condition. Recently in [18,19], the authors adapted discontinuous Galerkin methods to (1.1) and obtained the optimal H 1 error estimates. In this paper, we construct the fully discrete approximation of the problem (1.1) using the discontinuous Galerkin method for the spatial discretization and Crank-Nicolson method for the time step discretization.…”

A Crank-Nicolson Characteristic Finite Element Method for Sobolev Equations

“…For the existence, uniqueness, and regularity of the solutions of the Sobolev equation (1.1), we refer to [3,4,20]. For Sobolev equations without a convection term, many mathematicians achieve the numerical results by classical finite element methods [1,6,10,11,12] or least-squares methods [9,15,16,21,22] or mixed finite element methods [8] or discontinuous finite element methods [13,14,18,19]. But in many situations, the convection term d (x ) · ∇u exists and d (x ) is large in order to describe a convection dominated diffusion.…”