Du, Yanwei; Liu, Yang; Li, Hong; Tong, Mingwang A Crank-Nicolson splitting positive definite mixed finite element method based on two transformations for Sobolev equation with convection term. (English) Zbl 1324.65121 Adv. Math., Beijing 43, No. 6, 869-886 (2014). Summary: In this article, we propose and discuss a new splitting positive definite mixed finite element (SPDMFE) method for second-order Sobolev equation with convection term. We introduce two transformations: \(q=u_t\) and \(\sigma=a(x)\nabla u+b(x)\nabla u_t\) and solve the ordinary differential equation \(\sigma =a(x)\nabla u+b(x)\nabla u_t\) for \(\nabla u\), then reduce the Sobolev equation to a first-order integro-differential system with three variables. In the integro-differential system, the equation for the actual stress \(\sigma\) is independent, symmetric, positive definite, and can be solved independently from both the variable \(u\) and \(q=u_t\) then we can approximate the scalar unknown \(u\) and the variable \(q\). We derive a priori error estimate and stability for both semidiscrete and Crank-Nicolson fully discrete schemes. Finally, we provide some numerical results to illustrate the efficiency of the new SPDMFE method. MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 65R20 Numerical methods for integral equations 45J05 Integro-ordinary differential equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs Keywords:Sobolev equation; Crank-Nicolson scheme; transformation; error estimate; semidiscretization; splitting positive definite mixed finite element method; first-order integro-differential system; a priori error estimate; stability; numerical results PDFBibTeX XMLCite \textit{Y. Du} et al., Adv. Math., Beijing 43, No. 6, 869--886 (2014; Zbl 1324.65121)