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

In this paper,we will describe the mixed finite element method elliptic problems on non-matching multiblock grids,and 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 u=-K(▽p-γp) in Ω,(B.1) cp+▽·u=q in Ω,(B.2) u·v=0on (?)Ω.(B.3) where u and p represent velocity vector and pressure in physics,respectively. 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(?)Rd,where Ωf.Let Γi be the interior of (?)Ωi (?)Ω.Then we can define the inter-face between blocks i and j as Γij=Γi∩Γj for i≠j and Γii=0.The entire interface is Γ=Ui,jΓij.To conserve the continuity of p and u on the interfaces,the interface condition αpi-ui·vi=αpj+uj·vj onΓij,i,j=1,2…,n.(B.4) should be satisfied.Then the elliptic problem on non-matching multiblock grids can be represented as u=-K(▽p-γp) in Ωi,i=1,2,…,n,(B.5) cp+▽·u=q in Ωi,i=1,2,…,n,(B.6) u·v=0on (?)Ω,(B.7) αpi-ui·vi=αpj+uj·vj on Γij,i,j=1,2…,n.(B.8) Define function space H0(div; Ωi)={v∈(L2(Qi))d:▽·v∈L2(Ωi), v·vi=0on (?)Ω}.(B.9) Then the variational form of the problems (B.5)-(B.8) can be represented as:to find p∈L2(Ω), u∈(L2(Ω))d, such that pi∈L2(Γi), ui∈H0(÷;Ωi), for i=1,…n and (K-1u, v)i=(p,▽·v)i+(γp,v)i-<pi,v·vi>, v∈Ho(div;Ωi),(B.10){cp,w)i+(▽·u,w)i=(q, w)i, w∈L2(Ωi),(B.11) where (·,·),i denotes the norm on (L2(Ωi))d or L2(Ωi),<·,·>,i denotes the norm on L2(Γi),(·,·)ij denotes the norm on L2(Γij). Define the finite element space in Ω as Vh={v∈(L2(Ω))d:vi=v|Ωi∈Vh,i,i=1…,n),(B.13) Wh={w∈L2(Ω):wi=w|Ωi,∈Wh,i,i=1,…,n],(B.14) Ah={μi∈L2(Γi):μi,j=μi|Γi,j∈Λh,i,i=1,…,n}.(B.15) where Vh,i×Wh,i×Λh,i(?)H0(div;Ωi)×L2(Ωi)×L2(Γi)(B.16) Then the mixed finite element method on non-matching multiblock grids can be represented as finding uh∈Vh,Ph∈Wh,λh∈Γh such that(K-1uh,γhv)i=(ph,▽·γhv)i+(γph,γhv)i-<λh,i,γhv·v>i,v∈Vh,i (B.17)(cph,w)i＋(▽·uh,w)i＝(q,w)i,w∈Wh,i,(B.18) where uh,i=uh|Ωi,ph,i=ph|Ωi,λh,i=Ph|Γi.The existence and uniqueness of solution and the convergence are exists.Similar with the mixed finite element method on non-matching multiblock grids, with the interface condition (B.4), 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 Vh×Wh×Γh defined in(B.13)-(B.15),where Vh is chosen as the smallest order RT space,and Wh,Γh are chosen as piecewise constant spaces,namely,the trial function space is given by Vh,i={v:v|K∈Kh=(a+bx,c+dy),a,b,c,d∈R),(B.20) Wh,i={w:w|K∈Kh=constant},(B.21) Λh,i={μi:w|i∈Ii=constant}.(B.22) The dual subdivision is shown in Fig.0.2.Ki+1/2,j is called as u-block,Ki,j+1/2is called as v-block,Ki,j is concrete rectangular element.Then the corresponded test space is图0.2Rectangular original subdivision and dual subdivision defined as Vh×Wh×h,where Vh={v∈(L2(Ω))d:vi=v|Ωi∈Vh,i=1,…,n),(B.23) Vh,i={v:v|K∈Kh=(a+bx,c+dy),a,b,c,d∈R}.(B.24)Define the operator γh:Vh,i�'Vh,i,set uh=(uh,vh),then γhuh=(γhuh,γhvh) where, χi+1/2,j and χi,j+1/2represent the element Ki+1/2,j and Ki,j+1/2the characteristic func-tion. Then the mixed finite volume element method on non-matching multiblock grids can be represented as:solving uh∈Vh, ph∈Wh, λh such that (K-1uh, v)i=(ph,▽·v),+(｜Ph, v)i-<λh,i,v·v)i,v∈Vh,i,(B.26)(cph, w)i+(▽·uh, w)i=(q, w)i,w∈Wh,i,(B.27)Similarly, we also proposed the mixed finite element method and mixed finite vol-ume method on non-matching multiblock grids for Dirichlet boundary problem. We made numerical simulations for Neumann and Dirichlet boundary problems by using the mixed finite element method and mixed finite volume element method. The simula-tion result shows that the convergent rate of the two method is the same. Consequently, we proposed the mixed finite volume element method on non-matching multiblock grids which is more easier for computation while the convergent rate is reserved.