A split least-squares characteristic mixed finite element method for Sobolev equations with convection term (Q1037796)

For the linear equation in the title (ie. convection-diffusion with a term \(u_{xxt}\)), the authors consider a backward Euler approximation in time and mixed finite elements (for the solution and its flux) in space, the convection term being used to approximate the material derivative. All this then is put into a least squares formulation which can be split exactly (using partial integration to eliminate the mixed terms) into two independent parts. Stability and convergence of optimal accuracy (in \(L_2\) and \(H(\text{div})\), respectively) are proved and convincing numerical experiments in 1D are shown.