The Finite Volume Element Method For The Parabolic And Hyperbolic Equations

The Finite Volume Element Method, also called as Generalized Di?erence Method.Its primary merits include computational simplicity and preserving local conservationof certain physical quantities.In this paper, we consider finite volume element method for the parabolic andhyperbolic problems. First, under convex quadrilateral meshes and the assumption thatall elements on quadrilateral meshes are O(h~2) quasi-parallel quadrilateral elements.The optimal L~2 norm error estimates are obtained for the semi-discrete and full discretefinite volume element schemes of the parabolic equations, and the optimal H~1 and L~2norms error estimates are obtained for the semi-discrete finite volume element schemeof the hyperbolic equations, which also the error estimates are obtained for the twofull discrete schemes. The trial function space is taken as isoparametric bilinear finiteelement space, and the test function space is defined as piecewise constant space ondual partition. Second, based on triangulation and barycenter dual subdivisions, for aclass of variable coe?cient hyperbolic problems, the optimal H~1 and L~2 norms errorestimates are obtained for the semi-discrete symmetric finite volume element scheme,and the error estimates are also obtained for the two full discrete symmetric schemes.Finally, based on triangulation and BB dual subdivisions, for the variable coe?cienthyperbolic problems, the optimal H~1 and L~2 norms error estimates are obtained forthe semi-discrete finite volume element scheme, and the error estimates of the two fulldiscrete schemes are also obtained .