A Stable Mixed Finite Element Scheme for the Second Order Elliptic Problems

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1 A Stable Mixed Finite Element Scheme for the Second Order Elliptic Problems Miaochan Zhao, Hongbo Guan, Pei Yin Abstract A stable mixed finite element method (MFEM) for the second order elliptic problems, in which the scheme just satisfies the discrete B.B condition, is discussed in this paper. The uniqueness existence of solutions for the corresponding discrete problems are obtained, the optimal O(h 2 ) order error estimates are derived. Index Terms mixed finite element scheme, error estimates, elliptic problems equations. I. INTRODUCTION The mixed finite element methods (MFEM) are widely used in the theoretical analysis computations of many areas of science engineering. Because the method could approximate both the scalar variable its flux variable at the same time, obtain some convergence order, it becomes more more popular in the numerical computational areas. A lot of studies on MFEM have been devoted to the second order elliptic problems. For example, [] proposed a novel discontinuous MFEM, used the discontinuous piecewise polynomial finite element spaces for the trial test functions. [2] discussed an MFEM based on the Hellinger-Reissner variational principle, the maximum norm error estimates were obtained. [3] applied the MFEM to the 2D Burgers equation by using the P 2 0 P FE pair also obtained the corresponding optimal error estimates. [4] considered the a priori a posteriori error analyses of a mixed finite element method for Darcy s equations with porosity depending exponentially on the pressure by employing the local approximation properties of the Raviart- Thomas Clément interpolation operators as the main tools for proving the reliability. [5] developed a finite element mathematical model to study the flow of calcium in two dimensions with time, gave a lot of numerical experiments to illustrate the method. [6] applied the MEFM to numerically study the processes of compaction fluid flow in a sedimentary basin represented by a two-dimensional mode. [7] developed a nonconforming MFEM to solve the second-order elliptic problems on rectangular meshes, obtained the optimal order error estimate. Moreover, [4] applied the method of [7] to the elliptic optimal control Manuscript received January 7, 206; revised June 30, 206. This work was ported by the National Natural Science Foundation of China (Nos ), 206 General Project of Humanities Social Science Foundation of USST(6HJS-YB), 206 PhD Start-up Project of USST (BSQD20609), the Science Technology Projects of Henan province ( ). Miaochan Zhao is with the School of Mathematics Statistics, Zhoukou Normal University, Zhoukou 46600, China. Corresponding author. Hongbo Guan is with College of Mathematics Information Science, Zhengzhou University of Light Industry, Zhengzhou , China. guanhongbo@zzuli.edu.cn Corresponding author. Pei Yin is with Business School, University of Shanghai for Science Technology, Shanghai , China. pyin@usst.edu.cn problem. [9] investigated a mixed variational form for second elliptic problem automatically satisfies strongly elliptic condition the B.B condition, the optimal order error estimates were obtained. Recently, [0] constructed a low-order rectangular MFEM for second order elliptic problems, in which they used (Span, x, y, xy, x 2,Span, x, y, xy, y 2 ) for vector function Span, x, y for scalar function, the stability optimal order error estimates were derived. For the Variable-Coefficient elliptic problems, [] studied the multigrid block krylov subspace spectral method, some numerical results were shown to demonstrate the effectiveness of this approach. [2] introduced a weak Galerkin method for second order elliptic problems, the error estimates of H(div)-norm L 2 -norm for the primary flux variables are obtained, respectively, but the first mentioned is one order lower that the last one. In this paper, we will propose a nonconforming mixed finite element scheme for the same elliptic problem as above, obtain the optimal order error estimate. We use (Span, x, y, xy, x 2,Span, x, y, xy, y 2 ) for vector function Span, x, y for scalar function, in which the convergence orders of the primary flux variables are both O(h 2 ). The remainder of this paper is organized as follows. In Section 2, a new MFE scheme for elliptic problems is constructed. In Section 3, the corresponding mixed finite element space is obtained. In Section 4, we study the interpolation property consistency error of this MFE scheme, which shows that the optimal convergence estimate can be obtained. Throughout this paper, C is a generic positive constant independent of the mesh size h. II. ANALYSIS OF THE NEW MIXED VARIATIONAL SCHEME Consider the following elliptic problem: 2 ( a ij (x) u ) = f, in, x i,j= i x j () u = 0, on, is a rectangular domain in R 2, A = (a ij ) 2 2 is continuous, symmetry strongly elliptic on. Let ψ = A u, then divψ = f. The following equivalent mixed variational form of () is obtained, a(ψ, φ) + b(φ, u) = 0, ϕ H (2) b(ψ, v) = F (v), v M, a(ψ, φ) = A φ ψdx, b(φ, v) = vdivφdx,

2 F (v) = fvdx, H = ϕ ϕ L 2 () 2, divϕ L 2 (), M = v v L 2 (). Define the div-norm as follows: ϕ 2 H= ϕ 2 0, + divϕ 2 0,. (3) Let v = divφ in the second equation of (2), we have divψ divφdx = f divϕdx. (4) Substitution of the above equation into the first equation of (2) yields A φ ψdx udivφdx + divψ divφdx (5) = f divφdx Then, a new mixed variational scheme of () is obtained, it is to find (ψ, u) H M such that a (ψ, φ) + b(φ, u) = G(φ), φ H, (6) b(ψ, v) = F (v), v M, a (ψ, φ) = G(φ) = A φ ψdx + divψ divφdx f divφdx Lemma 2. a (, ) (H H) is a positive definite form, i.e., ψ H, there exists a positive number α > 0, such that a (ψ, ψ) α ψ 2 H Proof. Firstly, we verify that A satisfies the V - ellipticity, A = A = [ ] a22 a 2 a 2 a By the strong elliptic property of A, we know that there exists a positive number α such that (7) (8) a (x) > α > 0 (9) a (x)a 22 (x) a 2 2(x) > α > 0 (0) Obviously, a 22 0, then there exists another positive number β > 0 such that a 22 > β. () On the other h, considering that a a 22 (a 2 ) 2 > 0 a > 0, we have a 22 > 0. (2) Therefore, there exists a positive number α, such that a 22 > α > 0. (3) Because A is continuous strong-ellipticity on, there exists M > 0, such that A < M, which means that A > M. Noticing that a a 22 a 2 2 > α > 0, all of the order principal minor determinants of A are bigger than a positive number. Thus, A satisfies the V -elliptic property, a (ψ, ψ) = A ψ ψdx + divψ divψdx C (ψ 2 + divψ 2 )dx = C ψ 2 H, (4) The Lemma 2. is proved. Lemma 2.2 b(, ) (H M) satisfies the continuous B.B condition, i.e., there exists a positive number β > 0, such that v M, there holds Proof. Because b(ψ, v) β v M. (5) ψ H ψ H ψ 2 H = ψ divψ 2 0 ψ 2 0 +C ψ 2 C ψ 2, (6) then, for any given v L 2 (), let w H0 () H 2 () be the solution of the following equation: w = v in, w = 0 on. (7) We have w 2 C v 0 by using the regularity theory of partial differential equations, which refers to [3]. If we choose ( ) wx ψ 0 = H, w y then, Following the equation we have divψ 0 = v M. ψ 0 M C ψ 0 C w 2 C v 0, b(ψ, v) b(ψ 0, v) v 2 0 β v M, (8) ψ H ψ H ψ 0 H C v 0 The Lemma 2.2 is proved. Theorem 2. By using the results of Lemmas , the problem (6) has a unique solution (ψ, u) H M. III. CONSTRUCTION OF THE MIXED FINITE ELEMENT SPACE Let J h be a rectangular subdivision of, be a general element with four vertices (x e h, y e h 2 ), (x e + h, y e h 2 ), (x e + h, y e + h 2 ) (x e h, y e + h 2 ), F be an affine mapping from to, x = x e + h x, F : (9) x 2 = y e + h 2 x 2, is the rectangular reference element with four vertices (, ), (, ), (, ) (, ) in x x 2 plane.

3 Define the following functions on the reference element : ϕ ( x) = + x, ϕ2 ( x) = + x 2, (20) ϕ 3 ( x) = x, ϕ4 ( x) = x 2. We denote the four edges of by l i, on which ϕ i = 0 (i =, 2, 3, 4). Bi denotes the midpoint of l i, B,i = F ( B i ), l,i = F ( l i ), n,i (x) denotes the unit outside normal vector. ψ,i = n,i (B,i ) 4 ( φ j ) Q 2 (), which is zero j= i j on l,j (j i). Then, Q 2 () can be defined as: Q 2 () = Q () span ψ,,, ψ,4 = Q () Define ( + x )( x )( + x 2 ), ( + x )( x )( x 2 ), ( + x )( + x 2 )( x 2 ), ( x )( + x 2 )( x 2 ) = Q () x 2 x 2, x x 2 2, x 2, x 2 2, (2) X h = M h = ϕ h ϕh F vh P disc () Q 2 ( ), (22) be two conforming finite element spaces, P disc () is a nonuniform continuous piecewise one order polynomial space. The discrete form of (6) is to find (ψ h, u h ) (X h M h ), such that a (ψ h, φ h ) + b(φ h, u h ) = G(ψ h ), φ h X h, b(ψ h, ) = F ( ), M h. (23) For the convergence analysis in next section, we will give the definition of several subspaces. Xh, a subspace of X h, can be defined as X h = v C 0 () 2 v Q 2 (), J h, v = 0 X h. (24) We divide to R non-overlapping domains r (r =,, R), each one is Lipschitz continues open domain: = R r, r s = ϕ (r s). (25) r= Then, P is denoted by the first order polynomial spaces, define P () = v = v F v P, (26) Q h = P (J h ) = v L 2 () v P (), J h. (27) the following subspaces X h ( r ) : = v r v Xh, v 0 in \ r, (28) Q h ( r ) : = q r q Qh, (29) M h ( r ) : = Q h ( r ) L 2 ( r ), (30) M h : = q L 2 () q r = const, r =,, R. IV. CONVERGENCE ANALYSIS OF THE MIXED FINITE ELEMENT SCHEME (3) Firstly, based on the conforming finite element spaces defined in Section III, we have a (ψ h, ψ h ) = [ J h J J h J A ψ h ψ h dx + divψ h divψ h dx C (ψh 2 + divψh)dx 2 = C ψ h 2 H. (32) This is that a (, ) satisfy the strong ellipticity property on X h M h. Lemma 4. There exists a positive number λ > 0, the following B.B conditions holds on ( X h, M h ). λ q h 0, q h X h M h. (33) Proof. For any ϕ H 2 () 2, we introduce the operator Π h ϕ, Π h ϕ equals to ϕ at the four vertices, equals to the average value of ϕ on the four edges, Π h F Q 2. Then, M h, we have b(ϕ Π h ϕ, ) = = = J h div(ϕ Π h ϕ)dx div(ϕ Π h ϕ)dx ] (ϕ Π h ϕ) n ds = 0, Π h ϕ I h ϕ ϕ + ϕ ϕ + ϕ λ ϕ. (34) (35) By use of the Fortin principle [3], the B.B condition holds in ( X h, M h ). Lemma 4.2 There exists a positive number λ > 0, the following B.B conditions holds on r. λ q h 0,r, q h M h ( r ). X h ( r ),r (36) Proof. q M h (), we define v : R 2, v( x) := B T ( (q F k ))( x) b( x),

4 = (, ) T, x x b( x) = ( x 2 )( x 2 2). (37) 2 Obviously, b( x) > 0 in the, then v = 0 on the. Because of (q F ) P 2 k 2, we have v Q2 k. Noticing that b( x) v( x) = B T ( (q F ))( x)( b)( x) (38) = ( q)(f ( x))( b)( x) = b( x)b T ( q)( x) = b( x)( q)(f ( x)), (44) Considering that v = v F q = q F. We have X h(), then ( q)( x) = B T ( q)(f ( x)), (38) (divv, q) = (v, q) = v qdx = ( v)( x)( q)(f ( x)) detdf ( x) d x = (38) = h h 2 B T q( x) b( x)( q)(f ( x)) h h 2 d x b( x)( q) T (F ( x))( q)(f ( x))d x Next, we prove that there exist C > 0, such that b( x)( q) T (F ( x))( q)(f ( x)) detdf ( x) d x (39) C ( q) T (F ( x))( q)(f ( x)) detdf ( x) d x. (40) By use of (38), we get that ( q) T ( x)( q)( x) B k 2 = (B ( q) T (F ( x)), B T ( q)(f ( x))) B 2 ( q) T (F ( x))( q)(f ( x)) (4) B 2 ( q) T ( x)( q)( x) b( x)( q) T (F ( x))( q)(f ( x)) detdf ( x) d x ( q) T (F ( x))( q)(f ( x)) detdf ( x) d x b( x)( q) T ( x)( q)( x)d x B 2 B 2 ( q) T ( x)( q)( x)d x, (42) (divv, q) C ( q) T (F ( x))( q)(f ( x)) detdf ( x) d x C q 2, (43) v 2 0, = v T ( x) v( x) detdf k ( x) d x ( q) T (F ( x))( q)(f ( x)) detdf ( x) d x q 2,. (45) Next, we will obtain two positive numbers C 2 C 3, such that v, C 2 h v 0,, v X h () (46) q 0, C 3 h q,, q M h () (47) By using the result of (38), we have v 2, = B T ( v)(f ( x)) = B T ( v)( x). (48) 2 h h 2 B T ( v)( x) 2 detdf ( x) d x 2 ( v)( x) 2 d x. (49) By the equivalence property of the norms on the finite element spaces, there exists C such that v, C v 0,. On the other h, v 2 0, = v(x) 2 detdf ( x) d x v 2 0, 2h h 2. Thus the proof of equation (46) is completed. Similarly, q 2 0, = = C q 2 dx = q 2 det(df ( x))d x q 2 d x C meas(s ) C q 2, q 2 d x C meas(s ) [( q ) 2 + ( q ) 2 ]d x x x 2 [h 2 ( q ) 2 + h 2 x 2( q ) 2 ] dx x 2 h h 2 C q 2,, The proof of equation (47) is also completed. (50) (5) (52)

5 (46)(47) v, q 0, C 2 C 3 v 0, q, C 2 C 3 q 2 (43) (53), C 4 (divv, q) The B.B condition is satisfied. The following lemma given in [] shows the relationship between the global local B.B conditions: Lemma 4.3 The equations (33) (36) hold, there exists a positive β, such that β q h 0, q h M h. (54) X h Lemma 4.4 From Lemmas 4.3, b(, ) satisfy the B.B condition on X h M h, the problem (23) has a unique solution by the MFEM theory [3]. Finally, from Lemma , the optimal error estimate of this new mixed finite element scheme can be given as Theorem 2. Let (ψ, u) (ψ h, u h ) be the solutions of (6) (23), respectively, then we have the following error estimate: ψ ψ h H + u u h M Ch 2 ( ψ 3, + u 2, ). (55) REFERENCES [] F. Brezzi, J. Douglas L.D. Marini, Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik, vol. 47, no. 2, pp , 985. [2] L. Gastaldi R.H. Nochetto, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, RAIRO - Mathematical Modeling Numerical Analysis, vol. 23, no., pp , 989. [3] X.H. Hu, P.Z Huang X.L. Feng, A new mixed finite element method based on the Crank-Nicolson scheme for Burgers equation, Applications of Mathematics, vol. 6, no., pp , 206. [4] G. Gabriel, R.B. Ricardo T. Giordano, A mixed finite element method for Darcy s equations with pressure dependent porosity, Mathematics of Computation, vol. 85, no. 297, pp. 33, 206. [5] S.G. Tewari.R. Pardasani, Finite Element Model to Study Two Dimensional Unsteady State Cytosolic Calcium Diffusion in Presence of Excess Buffers, IAENG International Journal of Applied Mathematics, vol. 40, no. 3, pp. 08 2, 200. [6] E. ikinzon, Y. uznetsov, S. Maliassov P. Sumant, Modeling fluid flow compaction in sedimentary basins using mixed finite elements, Computational Geosciences, vol. 9, no. 2, pp , 205. [7] D.Y. Shi C.X. Wang, A new low-order non-conforming mixed finite-element scheme for second-order elliptic problems, International Journal of Computer Mathematics, vol. 88, no. 0, pp , 20. [8] H.B Guan, D.Y. Shi, A stable nonconforming mixed finite element scheme for elliptic optimal control problems, Computers & Mathematics with Applications, vol. 70, no. 3, pp , 205. [9] S.C. Chen, H.R. Chen, New mixed element schemes for second order elliptic problem, Mathematica Numerica Sinica, vol. 32, no. 2, pp , 200. [0] Y. im, H. Huh, New rectangular mixed finite element method for second-order elliptic problems, Applied Mathematics Computation, vol. 27, no. 2, pp , [] J. V. Lambers, A Multigrid Block rylov Subspace Spectral Method for Variable-Coefficient Elliptic PDE, IAENG International Journal of Applied Mathematics, vol. 39, no. 4, pp , [2] J.P. Wang, Y. Xiu, A weak Galerkin mixed finite element method for second order elliptic problems, Mathematics of Computation. vol. 83, no. 289, pp , 204. [3] S.C. Brenner L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 994. [4] X.F. Guan, M.X. Li, S.C. Chen, Some numerical quadrature schemes of a non-conforming quadrilateral finite element, Acta Mathematicae Applicatae Sinica. English Series., vol. 28, no., pp. 7 26, 202.