| Finite volume method is an important numerical methods developed from the finite difference method and the finite element method for solving partial differential equa-tions, which has advantages such as dealing with complex solving domain easily and keeping local conservation of physical quantities etc, and has been widely used in fluid dynamics, heat conduction equation, geology and other fields.1. Bicubic finite volume method on quadrilateral meshesConsider the following elliptic boundary value problem-▽-(A(x,y)▽u)=f, in Q,(1) u=0, on αΩ,(2)where Ω C R2is a bounded convex polygon with boundary αΩ, and f∈L2(Ω). The coefficient A(x, y)=(aij(x, y))i,j2=1, whose entries αij∈W1,∞(Ω), is assumed to be symmetric, bounded and uniformly positive definite in Ω; i.e., there exist two positive constants C1and C2such that C1ξTξ≤ξTA(x, y)ξ≤C2ξtξ,(?)ξ∈R2,(?)(x,y)∈Ω.(3)Divide Ω into a sum of finite number of strictly convex quadrilaterals such that different quadrilaterals have no common interior point, that a vertex of any quadrilateral does not lie on the interior of a side of any other quadrilateral, and that any vertex of the boundary is a vertex of some quadrilateral. Then we obtained a primal partition Th Let hK denote the diameter of the element K∈Th and ρK denote the maximum diameter of circles contained in K. Suppose Th is regular, i.e., there exist positive constants Cl and C2such that for any element K E Th hK/ρK≤C1,(4)|cosθK|≤C2<1, where θK is any interior angle of K.(5) A quadrilateral K is called an hl+T-parallelogram if it satisfies P1P2+P3P4|≤Ch1l+τ,(6)where0<τ≤1and Pi,i=1=1,2,3,4, are the vertices of the quadrilateral K. We take the unit square K=[0,1] x [0,1] as the reference element in the ξη-plane with vertices denoted byPI=(o, o), p2=(1, o), p3=(1,1), p4=(o,1) Let K be a convex quadrilateral with the Pi=(xi, yi)(Z=1,2,3,4) of the vertices placed counterclockwise. Then there exists a unique invertible bilinear transformation FK which maps K onto K and satisfies Pi=FK(Pi), i=1,2,3,4. On the reference element K, we select notes Pk,l=(σk,σl), for k, l=1,2,3,4, with σ1=0,σ2=1/2-(?)/10,e2=1/2,e3=1/2+(?)/10,σ4=1Welet Qij=(ei, ej), for i, j=0,1,2,3,4, with e0=0, e1=1/2-(?)/10,e2=1/2,e3=1/2+(?)/10,e4=1.For each element K E Th, let Pk,l=FK(Pk,l) and Qij=FK(Qi,j). We define the control volumes in K of the nodes Pk,l by the subregions Qk-1,l-1Qk,l-1Qk,lQk-l,l, where Qo,o=P1,1, Q4,o=P4,1, Q4,4=P4,4, and Qo,4=P1,4. For each node P, denote the associate control volume by KP. Then if P is the inner vertex, KP is the union of the above subregions, sharing the vertex P. If P is the node on one edge of some element, KP is the union of the above subregions, sharing the common edge including P. If P is the inner nodes of some element K, KP the above subregion, being part of K. The set of all the control volumes forms a dual partition Th.The trial function space Uh is chosen as a finite element space associated with the primal partition Th. Denote by Q3(K) the standard bicubic polynomial space on K, then defined the trial function space as Uh={uh∈H01(Ω):uh|K=uh o FK-1, uh∈Q3(K), K∈Th}.Define the test function space as Vh={vh∈L2(Ω): vh|KP*=constant, P∈Ωh|, where Qh denote the set of all the interpolation nodes.The finite volume element methods for (1) and (2) is to find uh E Uh such that a(uh, vh)=(.f, vh), vh∈Vh, with where P∈Ω and n is the outward unit normal vector of αKP*.Next, we proceed to the error analysis of the finite volume method. The following lemma gives the equivalence between the above discrete semi-norm and the associate continuous semi-norm.Lemma1. The norms|·|l,h and|·|are equivalent in Uh, i.e., there exist positive constants C1, C2independent of Uh such that C1|uh|l,h≤|uh|≤l≤C2|uh|l,h,(?)uh∈Uh.We now investigate the ellipticity of the bilinear form α(-, Ⅱh*),the proof of which relies on the following lemma about partitioned matrix. For each K∈Th with the average center Q, we denote AK=(α22(Q)-α12(Q)-α21(Q)α11(Q). Noting that AK is a real symmetry matrix, it has the factorization AK=QKAKQK, where AK is a diagonal matrix whose diagonal entries are the eigenvalues of AK, and QK is an orthogonal matrix. Due to the ellipticity condition (3), AK is positive definite. We now define an invertible mapping TKA by for x=(x, y)∈K, FKA(x)-QKTAK1/2QKx, and let KA:_TKA (K) and vector1-P,:_TKA (Pi), where P P is an edge of the element K. We define Th:={KA:KA=FKA (K),(?)K∈Th}. Regularity assumption KA. All the inner angles of all the elements in Th are uni-formly bounded away from zero and7, and the ratios between the length of the smallest edge and the longest edge of any element KA are uniformly bounded away from zero.Using Lemma1and Lemma2, we obtain the following theorem.Theorem1. Suppose that the partition Th satisfies the condition (6) and Regularity assumption KA. Then for sufficiently small h, there exists a positive constant C such that a(uh, ⅡH*uh)≥c‖uh‖12,uh∈UhUsing the ellipticity of the bilinear form a(-, IIh-), we can prove the following H1. error estimate.Theorem2. Suppose that u E Ho (Q) n H4(Q), uh is the solution to the FVEMscheme (7). If the conditions (4),(S) and (6), and Regularity assumption KA hold, then for sufficiently small h, there exists a positive constant C independent of h, such that‖u-uh‖≤Ch3‖u‖4. We apply the so called elementwise stiffness matrix analysis to the bilinear finite volume method and obtain the geometric requirements for ellipticity condition. Fur-ther, we compare the geometric requirements for ellipticity condition of finite volume methods, including bilinear, biquadratic and bicubic finite volume method schemes.Finally, we present some numerical examples to check the theoretical results of bicubic finite volume method on quadrilateral meshs.2. Adaptive bilinear element finite volume methods on nonmatch-ing gridsConsider the following model boundary-value problem:where Ω(?)R2is a bounded polygon with boundary αΩ,Γd(?)αΩ,Γn(?)αΩ,Γd∪T, αΩ and f E L2(Ω). For simplicity of description, we assume that the sides of S2are parallel to the coordinate axes. The length of fd is assumed to be positive to guarantee the uniqueness of the solution.Let Rh, a finite number Nh of rectangles Kjopen sets), j=1,2,…, Nh, denote a partition of the domain S2such thatNext we construct a dual partition Rh with respect to the primal partition Rh. Let εhi dnotes the union of all intersections of K E Rh and Eh the intersections of elements with the boundary OQ: εhi={e:e=K∩K’fo rsome K,K∈ERh}, εhi={e: e=K∩αΩ for some K∈Rh}, and set εh=εhi∪εhb.As the mesh Rh allows "hanging" vertices, we consider a subset of Eh, denoted by εh*.to formulate two different methods as following.Method1Let Eh be the set of those edges each of which contains at least one hanging vertex. Method2εh*=εh. For any element K∈Rh, we divide it into four sub-rectangles by connecting the mid-points of opposite edges. Denote the subregion corresponding to the vertex Pi E K by RK,Pi:=□MiQMi-1Pi, where i=1,2,3,4, and M0=M4. If Eh is chosen as in the Method1, denote by KP the control volume corresponding to the vertex P, which is the union of the above subregions associated to point P. To be more precise, KP*=∪K∈Rh,RK,All the control volumes form a dual partition Rh. If Eh is chosen as in Method2, we define the dual partition Rh to be the union of all the above subregions, e.g., Rh=JRK,P: VP E K, dK E Rhj.Based on the partition Rh, we define the trial function space Uh as the bilinear finite element space, Uh={uh∈C(Ω\εh*:u|k∈Q1(K),(?)K∈Rh,u|Γd=0}, where Q,(K) is the space of bi-polynomial of degree no more than1on K.The test function space Vh with respect to Rh is defined asVh={vh∈L2(Ω): vh|K’=constant,)(?)K∈Rh*, and vh uh|KP*0,(?)PΓd}. Let U(h)=Uh+(H2(Q) n Ho (Q)) and define a mapping IIh: U(h)-} Vh as rlhu J R,,,=u(Pi)-Define a bilinear form A: U(h) x U(h)-} R and define a functional on U(h) F(v)=(f, Ilhv)+∑e∈≈nfcgnⅡhvds+γ∑e∈Γdhe-1fcgdvds Finite volume methods for problem (1) are to find uh E Uh such that A(uh, vh)=F(vh), dvh E Uh.(9) Remark1. If εh*is chosen as in the Method1, we can derive a new finite volume method from formulation (9). In the new scheme, the trial function is continuous on the matching part of a grid and is discontinuous on the nonmatching part of it. If Eh is chosen as in the Method2, the formulation (9) is the discontinuous finite volume method. The discontinuity across the interelement edges is recovered by adding consistency and penalty terms as in discontinuous Galerkin method.2.1A priori estimationsLet Then The following equation about Al(-,-) holds(cf.[1]). Lemma3. For any v E U(h) and w E Uh, we have where7hv denotes the function satisfying Vhv I K=w I K for any K E IZh. Lemma4. There is a constant C independent of h such that Al(v, v)≥C|v|1,h2v∈Uh.The stability of the bilinear A(-,-) can now be proved. Lemma5. There exists a constant C>0independent of h such that for sufficiently large γ>0it holds A(v,v)≤C(?)v(?),(?)v∈Uh. Next we will show the bilinear A(-,-) is bounded.Lemma6. For any v, w E U(h), it holds A(v,w)≤C(?)v(?)(?)w(?), where C>0is a constant independent of h.Lemma7. Suppose that Eh\Eh is set of edges in Rh without hanging vertices. Then, for any∈H2(Ω), there exists a junction vh E Uh such that (?)u-vh(?)≤Ch(?)u(?)2, where C is independent of h and u.Combining Lemma6, Lemma5and Lemma7, we can obtain the following a priori error estimate.Theorem3. Suppose that εhi\εh*is set of edges in Rh without hanging vertices. u∈H2(Ω)∩Ho1(Ω) and uh∈Uh are the solutions to problems (8) and (9), respectively. For sufficiently large "y>0, there exists a positive constant C independent of h, such that (?)u-uh(?)≤Ch‖u‖2.By a standard duality argumetn,we hane the following error estimate in the L2norm.Theorem4.Suppose that εhi\εh*is set of deges in Rh without hanging vertices. u∈H2(Ω)∩01(Ω) and uh∈Uh are the solutions to problems (8)and (9),respectively. For sufficiently large γ>0,there exists a psoitive constant C independent of h,such that‖u-uh‖≤Ch2‖u‖2.2.1A posteriori error estimateTo give the a posteriori error estimator, we need to apply the following Lemma. Lemma8. For any υh∈Uh, there exists aχ∈Uh∩C(Ω) such that where C>0is a constant independent of h.We are now in a position to show the a posteriori error estimate.Theorem5. Suppose that u∈H2(Ω)∩H01(Ω) and uh∈Uh are the solutions to problems (8) and (9), respectively. Then, there exists a constant C>0independent of h such that|u-uh|1,h≤C·η, where η is given byNext, we will use the bubble functions to receive the efficiency bound of the a posteriori estimator. Define ωe=K1∪K2, where K1, K2∈Rh and K1∩K2=e. For K∈Rh, and e∈εhi, let φK and φe be respectively the element and edge bubble functions, which have the following properties(cf.[2]).Lemma9. Assume that the partition Rh is quasi-uniform. There exist the cut-off functions φK, φe∈C0(Ω) such that supp φK (?)K, supp φe (?)ωe,0≤φK≤1，0≤φe≤1, where C1, C2, C3, C4are positive constants independent of the discretization parame-ters. As in Refs.[3,4], if K(h),μ(h) and v (h) satisfy K (h)≤C(μ(h)+v (h)) and v(h)/μ(h)'0as h'0,then we refer to v (h) as higher order term (h. o. t.) compared with f (h). This concept will be adopted in the following theorem in order to simplify the notation.Theorem6. Assume that the partition7Zh is quasi-uniform. Let u∈H2(Ω)∩Ho (Ω) and uh∈Uh be the solutions to problems (8) and (9), respectively. Then, there exists a constant C>0independent of h such that hK‖f‖K≤C(|u-uh|1,K+h.o.t.),(?)K∈Rh, and he1/2(?)▽uh]e‖e≤C(|u-uh|1,we+h.o.t.),(?)e∈εhi.we present four numerical examples to demonstrate our methods in practice. The first example is to illustrate the convergence rate of numerical solutions on a series of uniformly refined grids containing hanging vertices. In the second example, we im-plement the Method1and Method2on the same nonmatching grid and compare their performances. The third and fourth examples are chosen to validate the adaptive finite volume methods and demonstrate the efficiency of the a posteriori error estimator de-fined by (10). |
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