A. D. P. I. approximation for characteristic finite element methods for quasilinear Sobolev equations and its analysis (Q2746538)

This paper is concerned with the characteristic finite element approximation for the quasilinear Sobolev equation: NEWLINE\[NEWLINEc(x,t) u_t +\vec{d}(x,t)\cdot\nabla u -\nabla\cdot[ a(x,u)\nabla u + b(x,u) \nabla u_t]=f(x,t,u),\quad (x,t)\in\Omega\times (0,T)\tag \(*\) NEWLINE\]NEWLINE together with initial and periodic boundary conditions, where \(\Omega =(0,1)\times (0,1)\), \(a,b,c,\vec{d}\) and \(f\) are given smooth functions of their arguments. The paper mentions that the problem (\(*\)) arises from the theory of seepage of homogeneous liquids in fissures rocks. NEWLINENEWLINENEWLINECharacteristic finite element methods have been applied to convection-dominated diffusion problems and alternating direction Galerkin methods to parabolic and hyperbolic problems. This paper presents an alternating direction preconditioned iteration (A.D.P.I.) for the characteristic finite element approximation for problem (\(*\)) by solving approximately the linear equations arising at each time step in a characteristic finite element method. Using the Sobolev projection of solutions to problem (\(*\)) and an estimate involving the projection in the \(H^{-1}\)-norm, the author obtains optimal \(L^2\)-error estimate for the iteration. Moreover, the quasi-optimal computational costs are given.