A fully discrete H¹-Galerkin method with quadrature for nonlinear advection-diffusion-reaction equations

We propose and analyze a fully discrete H¹-Galerkin method with quadrature for nonlinear parabolic advection-diffusion-reaction equations that requires only linear algebraic solvers. Our scheme applied to the special case heat equation is a fully discrete quadrature version of the least-squares method. We prove second order convergence in time and optimal H¹ convergence in space for the computer implementable method. The results of numerical computations demonstrate optimal order convergence of scheme in H ^k for k = 0, 1, 2.