Mixed finite volume methods on quadrilateral grids for second-order elliptic problems

Abstract

In this thesis we construct and analyze two kinds of mixed finite volume methods on general quadrilateral grids in which finite volume methodology is applied to the mixed formulation of a second-order elliptic problem. The first one is the mixed covolume method in which two staggered grids are utilized to define the control volumes around the unknowns of variables. It was designed by Russell who called it control-volume mixed finite element method and tested it for various problems on rectangular and quadrilateral grids. he first rigorous error analysis was provided by Chou and Kwak who reformulated it as their mixed covolume method and applied the covolume methodology. However, their treatment was restricted to rectangular grids, and a different approach should be taken to create the locally supported test functions for quadrilateral grids. The approach proposed here is based on the idea of mapping the unit coordinate vectors (which are natural test functions) on the reference element under the Piola transformation. With these new test functions the covolume methodology above can be applied again to establish optimal error estimates for the new mixed covolume method. It will be also shown that Russell`s scheme is a variant of this new method with a proper use of quadrature rules, and hence our analysis covers Russell`s quadrilateral case as well. We also apply the mixed covolume method to quasi-linear second-order elliptic problems. For error analysis we follow the argument by Milner who considered the mixed finite element method for the same problem. In doing so, we need to adapt the duality argument of Douglas and Roberts to the mixed covolume method. The second kind of mixed finite volume method analyzed in this thesis is the finite volume box method introduced by Courbet and Croisille who applied the idea of Keller`s box scheme to Poisson`s equation on triangular grids. We extend their method to general tensor coefficients and quadrilateral...