L~2 Error Estimates And Superconvergence Of The Finite Volume Element Methods On Quadrilateral Meshes

The Finite Volume Element Method, also called as Generalized Di?erence Method,was firstly put up by professor Li Ronghua in 1982. Its computational simplicity andpreserving local conservation of certain physical quantities,make it be widely used incomputing ?uid mechanics and electromagnetic field and other fields.In this paper, we consid?er the following Poisson equation:where ? ? R2 is a bounded convex polygon with boundaryΓ= ??, and f∈L2(?).The corresponding variational problem for (0-1) is: find u∈U := H01(?) satisfyingwhereDivide ? into a convex quadrilateral partition Th. Suppose that Th is quasi-uniform, i.e., there exist positive constants C1 ,C2 such thatSuppose that the following regular condition be satisfied for any quadrilateral K∈Th:(i) There exist constants C1,C2 > 0 independent of h, such that(ii) For any interior angleθof K∈Th, there exist constantsθ0 > 0 independentof h, such thatWe take the unit square K = [0,1]×[0,1] on the (ξ,η) plane as a reference element.For any convex quadrilateral K, there exist a unique invertible bilinear transformationFK which maps K onto K.Under the above conditions, we obtain the following results: I. The Optimal L2 Error Estimates For The Isoparametric Bilinear FiniteVolume Element MethodsThe trial function space Uh is taken as isoparametric bilinear finite element spaceon quadrilateral partition, and the test function space Vh is defined as piecewise con-stant space on dual partition. So, the finite volume method for (0-1) is : find uh∈Uhsuch thatwhereWe call that quadrilateral element K = P1P2P3P4 satisfies quasi-parallel quadrilat-eral condition, ifIn fact, we give two methods to prove the optimal L2 error estimats.i. The First MethodIntroduce an auxiliary problem: for ?g∈L2(?), find w∈H01(?) such thatWe assume that (0-2) and (0-8) are regular, i.e., for any f,g∈L2(?), the correspondingsolution u = uf and w = wg are in the space H01(?)∩H2(?), and there exists a constantC such thatLemma 0.1 It holds that(g,u?uh) = a(u?uh,w?Πhw)+(f,Πhw?Π?hw)+ah(uh,Πh?w)?a(uh,Πhw). (0-10)Noticing that |a(u ? uh,w ?Πhw)|≤Ch2||u||2||w||2≤Ch2||f||0||g||0, we needonly to estimate |(f,Πhw ?Πh?w)| and |ah(uh,Πh?w) ? a(uh,Πhw)|.Lemma 0.2 Suppose that (0-3), (0-4), (0-5) and (0-7) are satisfied, and f∈H1(?), then there exist constant C independent of h such that Write ah(uh,Πh?w) ? a(uh,Πhw) = E1 + E2, whereIn the sequel, we estimate E1 and E2 element by element. We have E1 = 0when K is a parallelogram, because ??2xu2h and ??2yu2h are both constants, and K(Πhw ?Πh?w)dxdy = 0. And for E2, we calculate the integrals on the boundaries of everyelement exactly, and combine the integrals on opposite edges and estimate them. Withthe quasi-parallel quadrilateral condition (0-7), we obtain |E1|≤Ch2||f||0||g||0,|E2|≤Ch2||f||0||g||0, respectively. Moreover, we conjecture that the su?cient condition (0-7)can not be weakened any more.Lemma 0.3 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5) and(0-7) are satisfied, and f∈L2(?), thenLemma 0.4 Under assumptions of Lemma 0.3, we haveWith the above four lemmas, we have the following L2 estimate:Theorem 0.1 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5)and (0-7) are satisfied, and let u∈H01(?)∩H2(?) be the solution to (0-1), anduh∈Uh to (0-6), then for any f∈H1(?), we haveii. The Second MethodNotice thatwe need only to estimate |a(u ? uh,Πhw) ? ah(u ? uh,Πh?w)|. Theorem 0.2 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5)and (0-7) are satisfied, and let u∈H01(?)∩H3(?) be the solution to (0-1), anduh∈Uh to (0-6), then we haveII. The Superconvergence For The Isoparametric Bilinear Finite VolumeElement MethodsAs we know, in the error analysis, the determination of the upper bound of ||u ?uh||m is usually reduced to the estimation of ||u ?Πhu||m and a(u ?Πhu,Πh?wh). Bythe approximation theory, in general we can only obtain, limited by degree k of theapproximate polynomials, thatIn general this estimate can not be improved even if the solution u possesses an highersmoothness. Therefore,is the optimal order error estimate. But this face does not exclude the possibility thatthe approximation of the derivatives may be of higher order accuracy at some specialpoints, called optimal stress points.Just as in the case of finite element methods, we can also obtain superconvergenceresults for finite volume methods, provided we manage to get the super interpolationweak estimates. In detail we have the following theorem.Lemma 0.5 Let u and uh be solutions of the boundary value problem andits finite volume element scheme, respectively. Assume the bilinear form of the finitevolume element scheme satisfies the following interpolation weak estimate: There existsp∈[1,∞] such that then one hasMoreover, let Nk be the set of interpolation optimal stress points: For any P∈Nk,there exists q∈[1,+∞] such thatThen one has ?where r is the number of points in Nk.To get the superconvergence, the partition Th has to satisfy the h2 uniform con-dition, which also be used in the finite element methods.Call that the quadrilateral partition Th is h2 uniform, if any two h2 quasi-parallelquadrilateral elements K1 = P1P2P3P4, K2 = P4P3P5P6, which share the commonedge P3P4, satisfyUnder the h2 uniform condition, we prove the first weak estimate of interpola-tion to 2 order, further more, we get the superconvergence of the solutions of thefinite volume element methods at three type points, at which the interpolation possessuperconvergence under the average gradient norm.Theorem 0.3 Suppose that Th is a h2 uniform quadrilateral partition, let u∈H01(?)∩H3(?)∩W2,∞(?), thenTheorem 0.4 Suppose that Th is a h2 uniform quadrilateral partition, let u∈H01(?)∩H3(?)∩W2,∞(?) be the solution to (0-1), uh∈Uh to (0-6), then consequentlywhere Si, i = 1,2,3 denote three types of set of optimal of stresses for Uh-interpolation,i.e. the centers of the elements, the inner vertices and the midpoints of inner edges,and denotes the average value of gradient , ri is the number of points in Si.III. New Formula And Convergence Analysis Of Isoparametric Bi-quadratic Finite Volume Element MethodsFor the Lagrange type finite element space, besides the isoparametric bilinearelements, the isoparametric biquadratic elements are also used commonly. The quarterpoints and three-quarter points of edges of one quadrilateral are taken as the nodes ofthe dual partition in existing formula. It seems reasonable to choose this, because heformula guarantee 2 order convergence in H1 norm. However, they are not the optimalscheme, since the convergence order is only 2 in L2 norm. We choose a new plan toget dual partition, construct a new FVEM formula, and prove the optimal H1 and L2error estimates. Moreover, we present detailed numerical experiments.For any K∈Th, with vertices Pi,i = 1,2,3,4, taking points Pij, such thatThen the control volumes in K of a vertex Pi, a midpoint Mi and averaging center Qare the subregions PiPi(i+1)PiPi(i?1), Pi(i+1)P(i+1)iPi+1Pi and P1P2P3P4 respectively.Then for each vertex P, we associate a control volume KP?, which is built by theunion of the above subregions, sharing the vertex P. For each midpoint M on anelement edge, we associate a control volume KM?, which is built by the union of theabove subregions, sharing a common edge including M. The set of all the controlvolumes makes the dual partition Th ? .The trial function space Uh is taken as isoparametric biquadratic finite elementspace on quadrilateral partition, and the test function space Vh is defined as piecewiseconstant space on dual partition. The finite volume method for (0-1) is : find uh∈Uhsuch that whereFirstly, we obtain the coercivity of the bilinear form.Lemma 0.6 Suppose that every element in Th satisfies the regular conditionand h2 quasi-parallel quadrilateral condition. Then for su?ciently small h, there exista constant C0 > 0 such thatFurther more, we prove the H1 error estimate.Theorem 0.5 Suppose that u∈H01(?)∩H3(?) is the solution to equation (0-1), uh is the solution to the finite volume formula (y:2:1.2.9), If conditions (0-4), (0-5)and (0-7) hold, then there exist a positive constant C independent of h such thatWe also get the optimal L2 error estimates.Theorem 0.6 For quadrilateral partition Th, suppose that (0-3), (0-4), (0-5)and (0-7) are satisfied. Let u∈H01(?)∩H4(?) and uh∈Uh are the solutions to (0-1)and(0-6) respectively, thenIV. Superconvergence Analysis Of Isoparametric Biquadratic Finite Vol-ume Element Methods on rectangular meshesAs we know, the four Gauss points in every quadrilateral are the optimal pointsof stresses for isoparametric biquadratic Uh-interpolation. the solutions of the finiteelement methods are superconvergent at these optimal points of stresses. In this paper,we prove that the derivative of the solutions for the isoparametric biquadratic FVEMare also superconvergent at these points. Numerical experiments show that the super-convergence also holds on quadrilateral meshes. But we can't prove it with the methodused on rectangular meshes. Firstly, we introduce Legendre orthogonal polynomials on E = (?1,1)When i = j, we have (li,lj) = 0, and (lj,lj) = 2j2+1, lj(±1) = (±1)j. We obtain theM type polynomials from the integral on ljObviously, if n 2, Mn(±1) = 0. And if j ? i = 0 or±2, (Mi,Mj) = 0, (Mi,Mj) = 0otherwise.To study the superconvergence of the finite volume methods for 2nd order ellipticequations, we will construct the M type expansion polynomials on square K = [?1,1]×[?1,1]. Suppose that u is smooth enough. Fix t, expand u on s to obtainthen for every bi(t), we take M type expand on twith coe?cientsSo, we can obtain the M type expansion for u with these coe?cientsLemma 0.7 Suppose u∈W4,p(?), 1 p∞, then we have the following estimates of the interpolation uIwhere, Z is the set of the Gauss points in element.Theorem 0.7 Suppose that Th is a regular rectangular partition, let u∈H01(?)∩H4(?), and uI be the orthogonal interpolation polynomial of u, thenWe have the following superclose estimate:Theorem 0.8 Let Th be a regular rectangular partition, If u∈H01(?)∩H4(?)and uh∈Uh are the solutions to (0-1) and (0-18) respectively, thenfurtherWhere S is the set of the optimal points of stresses for Uh interpolation, i.e. the Gausspoints in element. denotes the average value of gradient , r is the number of pointsin S.