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A Crank-Nicolson splitting positive definite mixed finite element method based on two transformations for Sobolev equation with convection term. (English) Zbl 1324.65121

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
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