The Mixed Finite Volume Element Method For Elliptic Problems On Non-matching Multiblock Triangular Grids

In this paper, we will describe the mixed finite element method for ellip-tic problems on non-matching multiblock grids, and on this method, we will also propose the mixed finite volume element method on non-matching multiblock grids. The Neumann boundary problem of elliptic equations can be represented as where u and p represent velocity vector and pressure in physics, respective-ly.As the properties of the media in different regions of ? is different, then the equations above should be solved in different regions. Consequently, the elliptic problems on non-matching multiblock grids should be studied. Let the open set ? be decomposed into non-overlapping subdomain blocks ?i, i=1,…,n; that is,? is the interior of Ui=1n?i(?)R2, where ?i. Let ?i be the interior of (?)?i (?)?. Then we can define the interface between blocksTo conserve the continuity of p and u on the interfaces, we have the interface conditionBy using this interface condition, the elliptic problem on non-matching multiblock grids can be represented as Define function spacesThen the variational form of the problems can be represented as:find-where (·,·)i denotes the norm on (L2(?i))2 or L2(?i),<·,·>i denotes the norm on L2(?i),<·,·>)ij denotes the norm on L2(?ij). Define the finite element space in ? asThen the mixed finite element method on non-matching multiblock grids can be represented as:finding uh?Vh, ph?Wh, ?h??h such that where uh,i=uh|?i,ph,i=ph|?,?h,i=ph|?i, The existence and uniqueness of solution and the convergence have been proved(literature [30]).Similar with the mixed finite element method on non-matching multi-block grids, with the interface condition (27), we proposed the the mixed finite volume element method on non-matching multiblock grids. During the finite volume element method, the trial function space is chosen as Uh×Wh×?h defined in (36)-(38), where Uh is chosen as the smallest order RT space, and Wh, ?h are chosen as piecewise constant spaces, namely, the trial function space is given byThe dual subdivision is shown in Fig.2. where Ki represent the ith tri-angle, ?ik,k=1,2,3, respectively represent the kth sides of Ki, and nik, k= 1,2,3, respectively represent the unit normal vectors of the kth sides of Ki, and Til, l=1,2,3, respectively represent the lth dual element of Ki. Then the corresponded test space is defined as Vh×Wh×Ah:Definite a projection operator from the trial function space to test func-tion space ?h:Uh?Vh by where |?ik| represent the length of ?ik. Obviously the projection operator ?h establishes a one-to-one mapping from Uh to Vh.Then the mixed finite volume element method on non-matching multi-block grids can be represented as:finding uh?Uh,ph?Wk,?h??h forThrough numerical experiments, the results show that the mixed finite volume element on non-matching multiblock triangular grids and the mixed finite volume element method have the same convergence. So we can know the mixed finite volume element method on non-matching multiblock trian-gular grids is correct.