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 Ω 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 Ω be decomposed into non-overlapping subdomain blocks Ωi, i=1,…,n; that is,Ω is the interior of Ui=1nΩi(?)R2, where Ωi. Let Γi be the interior of (?)Ωi (?)Ω. 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(Ωi))2 or L2(Ωi),<·,·>i denotes the norm on L2(Γi),<·,·>)ij denotes the norm on L2(Γij). Define the finite element space in Ω asThen the mixed finite element method on non-matching multiblock grids can be represented as:finding uh∈Vh, ph∈Wh, λh∈Γh such that where uh,i=uh|Ωi,ph,i=ph|Ω,λh,i=ph|Γi, 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×Γh 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, Γik,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 γh:Uh→Vh by where |Γik| represent the length of Γik. Obviously the projection operator γh 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,λh∈∧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.