The Finite Volume Element Method Based On Circumcenter Dual Subdivision

In this paper, we obtain the optimal L2 error estimates of the finite volume element method based on circumcenter dual subdivision for the elliptic equation, and also prove the optimal L2 and Hl error estimates of the semi-discrete and fully-discrete finite volume element method based on circumcenter dual subdivision for parabolic equation.Firstly consider the second order elliptic boundary value problem:let R2 be polygonal region with boundary , where the coefficients i,j are suffieciently smooth functions satisfying the elliptic condition, we also require f L2( ) Let Thbe triangular subdivision of , Th*be circumenter dual subdivision. The trial function spaceUh is chosen as the linear element space related to Th;The test function spaceVh is chosen as the piece-wise constant function space respect to Th*, spanned by the following basis functions: For any point P0 K.For any vh Vh, vh =The volume element method for(l) is:Find uh Uh,such thatwhereand where n is the unit outer normal vector andTherorem 1 Let u be the generalized solution to (1), and uh the solution to (2), assume the distances between the barycenter Q and circumcenter C of any triange element in Th satisfy |QC| = O(h2}, then the following error estimate holds:Secondly consider the parabolic differential equation:Then the semi-discrete finite volume element method for(6) is: Find uh = uh( ,t) Uh(0 t T), such thatwhere u0h Uh is a certain approximation ofquotient tuhn = (uhn - uhn-1)/ to approximate the differential quotient Uh,t, then we obtain a fully-discrete backward Euler finite volume element scheme: Find uhn Uh(n = 1,2,..., M), such thatIf we discretize the semi-discrete scheme at time tn-1/2 = (n-1/2)rina symmetric fashion, then we have another fully-discrete Crank-Nicolson finite volume element scheme: Find uhn Uh (n = 1,2,..., M), such thatTherorem 2 Let u and uh be the solution to (6)and semi-discrete finite volume element scheme(7)respectively, thenTherorem 3 Letit and {uhn} be the solutions to (6) and the backward Euler finite volume element scheme (8) respectively, thenTherorem 4 Letu and {uhn} be the solutions to (6) and the backward Euler finite volume element Crank-Nicolson scheme(9) re-spectively ,then...