The Biquadratic Weighted Upwinding Finite Volume Method On The Rectangular Grids

This paper mainly focuses on the weighted upwind biquadratic finite volume method for the convection diffusion problem.The difference between the weighted upwind finite volume method and the standard upwind finite volume method is that the convection term in the line integral is usually approxiamated by the pure upstream value.While in the weighted upwind finite volume format is mainly determined by the new dual mesh.which depends on thePecletnumber.The weighted upwind biquadratic finite volume method keeps the stability of the standard upwind finite volume style,with the optimal L2 convergence rateIn order to construct a weighted upwind finite volume format,we first make a rect-angular partition Th of the domain,take the biquadratic finite element space Uh corre-sponding to the original partition as the trial function space.Define the Peclect num-ber,which relies on the mesh size,the convection parameter and the diffusion parame-ter.And,eonstruct the dual mesh Th*according to a function of the Peclect number.Then the piece-wise constant test space Ve is chosen based on the dual mesh Th*.The weight-ed upwind finite volume format for solving the convection-diffusion equation is:Find Uh ? Uh such that a(uh,vh)+b(uh,vh)=(f,vh),(?)vh?Vh.Where,#12 Then,we proved the stability of the format:#12 and obtained the H1 norm error estimate:|u-uh|1?Ch2|u|3.Finally,The numerical experiment proves that the order of H1 norm convergence of the weighted upwind biquadratic finite volume method is |u-uh|1=O(h2),And the order of L2 norm convergence is ?u-uh?0=O(h3),the proof of the L2 norm error estimate will be studied in the future.