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Galerkin Continuous Finite Element Methods for a Forward-Backward Heat Equation Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221 Donald A. French July 8, 1998 Abstract f = f(x; t) such that Consider the following forward-backward initial/boundary value problem: nd u = u(x; t) given 1. Introduction: A space-time nite element method is introduced to solve a model forward-backward heat equation. The scheme uses the continuous Galerkin method for the time discretization. An error analysis for the method is presented. with boundary conditions (BC) xut uxx = f in = ( 1; 1) (0; 1) u = 0 on u = 0 on + 0 + 1; (1) (2) (3) and initial/post conditions (IC/PC) where + = f(1; t) : t 2 (0; 1)g; = f( 1; t) : t 2 (0; 1)g; + = fx : x 2 (0; 1)g; = fx : x 2 ( 1; 0)g; and, in particular 0 is at t = 0, and 1 is at t = 1. We will often use the notation = ( 1; 1). + + This equation arises in electron scattering (see B]), uid ow near boundary layers (see GM]), and random acceleration of a particle (see FR]) (see also the discussion in AFJK] for a full background). There have been several studies of the partial di erential equation (see BG] as Research supported in part by the Taft Foundation at the University of Cincinnati through their Grants-in-aid. 1 well as the other papers listed in the references on this problem). Numerical work includes the investigation of a nite di erence scheme in VK] and methods using the transformation of (1) to a rst order system were carried out in AL1]. A least squares approach is described in AL2]. In AFJK] a two-dimensional nite element scheme for (1)-(3) is analyzed and in F] a discontinuous Galerkin(dG) nite element scheme is studied. The continuous Galerkin(cG) time stepping scheme which we will study is closely related to the Gauss-Legendre Implicit Runge-Kutta schemes (IRK) (see AM] and FS] for more on cG methods and BDK] for an application of IRK schemes to the KdV equation). The cG scheme can achieve the same high accuracy as the IRK scheme (see FP]) and because of its variational structure is amenable to the development of a posteriori error estimates that are useful in adaptive computations (see EF]). Closely related to the cG method is the dG method (see T]). In an ordinary di erential equation setting the cG method can achieve convergence rates proportional to k2m at the nodes if there are m degrees of freedom on each step with length k while the dG method will achieve slightly lower rates that are proportional to k2m 1 . This is nicely exempli ed in the m = 1 cases. The cG method is closely related to the Crank-Nicholson method (O(k2 )) while the dG method is related to the backward Euler scheme (O(k)). Our main result is on a uniform grid where the re nements by parameter h > 0 in space and time are identical and the approximation spaces consist of piecewise polynomials of degree p. We show that ZZ 1=2 Chp+1 (4) x2e(x; t)2 dxdt where e is the error. Our general theorem will also include the case where space and time re nements could be di erent as could the degrees of the piecewise polynomial functions. A weak formulation which is used to motivate the nite element scheme can be found by 1 multiplying (1) by v 2 H0 ( ) and integrating over . The set H m (A) for some domain A is the 1 Sobolev space with functions that have m derivatives in L2 (A). The space H0 (A) is functions in H 1(A) which have trace zero on the @A. Integration-by-parts and the BC are used to rewrite the term involving uxx , 1 (xut ; v) + (ux ; vx ) = (f; v) ; 8v 2 H0 ( ); (5) 1 where the desired solution u 2 H0 ( ) and for a domain A we de ne the L2 inner product (w; v)A = By setting v = u we have Z A wvdA and norm jjwjj2 = (w; w)A : A 1 d (xu; u) + ku k2 = (f; u) : x 2 dt 2 De ning we obtain, using the inequality with = 1, that kwk 1; = max (w; v) 1 v2H0 ( ) kvx k (6) 1 ab 2 a2 + 2 b2 Integration with respect to t over 0; 1]gives d (xu2 ; 1) + ku k2 x dt 1=2 uk2 kfk2 1; : 2 kfk2 1; dt: (7) + kux k 0 This is a fundamental a priori estimate for the problem and one we will exploit in our analysis. The estimate (7) shows that if (1)-(3) has a solution it will be unique. We will assume throughout that there is a unique solution which is as regular as required by an analysis. We will use the Poincare inequality; kuk Ckuxk for u 2 H01 ( ) (8) in the succeeding sections. kx 1=2 uk2 1 + kjxj + 0 Z1 In this section we introduce the space-time nite element scheme that uses the cG method for the time discretization. We rst specify the grids. Let h = 1=(M + 1) and xj = jh where j = 0; 1; 2; : : : ; (M + 1). Let k = 1=N and tj = jk for j = 0; 1; : : : ; N. We now turn to the de nition of the approximation spaces. We will keep careful track of p dimension to ensure that the resulting problem leads to a square system of equations. Let Xh be the space of continuous piecewise polynomials of degree p on the spatial grid which are zero at 1. Subtracting the continuity and boundary condition constraints from the degrees of freedom q p we nd that dim Xh = (2M + 2)p 1. For the temporal discretization we let Ck be the set of r continuous piecewise polynomials of degree q and Dk be piecewise polynomials, not necessarily q continuous, of degree r. Counting the constraints of Ck and degrees of freedom for each we nd q = Nq + 1 and dim Dr = N(r + 1). that dim Ck k q p The solution space Shk will consist of all functions in Xh Ck which are zero at all nodes on 1 and all nodes on 0 except (0; 0). Thus + p q dim Shk = (dim Xh ) (dim Ck ) fIC and PC constraintsg = (2M + 2)p 1] Nq + 1] (2M + 2)p 1] = (2M + 2)p 1]Nq: 3 2. Approximation Method: q ( t v; ) = (v; ) 8 2 Dk 1 ; q where = (0; 1). De ne Pt : H 1 ( ) ! Ck by q ((Pt v)t ; t ) = (vt ; t ) 8 2 Ck with Pt v(0) = v(0) and it then follows that Pt v(tj ) = v(tj ) for j = 1; 2; : : : ; N. Finally, de ne p 1 Px : H0 ( ) ! Xh by p ((Px v)x ; x ) = (vx ; x ) 8 2 Xh p and x : L2 ( ) ! Xh by p (( x v); ) = (v; ) 8 2 Xh : Note, in particular, that t and Pt act independently of the spatial operators, inner products and functions while Px and x act independently of the temporal operators, inner products, and functions. We now proceed to give the approximation properties for the spaces and operators which we will use. See Ciarlet C] for discussion and justi cation of these standard results. We will use the following inverse inequalities: for 2 Pr (In ) With these notations the cG method is de ned by the following variational problem: nd U 2 Shk such that (xUt ; ) + (Ux ; x ) = (f; ) 8 2 Thk : (9) q We will use several projection operators throughout. De ne t : L2 ( ) ! Dk 1 by p q The test space is Thk = Xh Dk 1 and p q dim Thk = (dim Xh ) (dim Dk 1 ) = (2M + 2)p 1]Nq = dim Shk : k kL1 (In) Ck 1=2 k kIn and k t kIn Ck 1k kIn ; p where In = (tn 1 ; tn ). For 2 Xh we have k xk d Ch 1 k k and k( dx )` kL1 ( ) (10) (11) Ch 1=2 We now summarize the key properties we will use for the operators x ; t ; Px ; and Pt . For v 2 H s ( ) there is a constant C such that d k( dx )` k ; ` = 0; 1: k(I Pt )vk Ckskvks; and k(I t)vk Ckr kvkr; ; 0 r; s q + 1: (12) The measure, k k`; is the H ` ( ) norm. For the spatial operators we have k(I Px )wk`; Chs `kwks; and k(I x )wk`; Chr `kwkr; ; 0 r p+1; ` = 0; 1: (13) 4 We note that, in particular, Px is bounded in L2 ( ). It can also be shown using Px , the inverse estimate (11), and the approximation properties (13) that k xwk1;Sn Ckwk1;Sn 8w 2 H01 ; ( ) (14) where Sn = In . The following lemma gives a useful bound involving t . Denote evaluation at time level tn with a superscript n. Lemma 1: For any real number 0 there exists a constant C such that k jxj V k2 k we have N X n=0 q p k jxj V nk2 + Ck jxj tV k2 8V 2 Xh Ck : (15) Proof: Adding the weight function jxj to the proof in French and Jensen ( FJ] lemma 1, p. 427) k jxj V k2 n kk jxj V n 1k2 + Ck jxj tV k2 n S S N X n=1 and summing from 1 to N we have k jxj V k2 k k jxj V n 1k2 + Ck jxj t V k2 3. Preliminary Estimates: and from this (15) follows. 2 Finally, we note that the following trivial weight function inequality holds for any real number 0: k jxj vk kvk : (16) In this section we derive several estimates from the cG variational equation: (xVt ; )Sn + (Vx ; x )Sn ; = (g; )Sn (17) where V 2 Shk , 2 Thk , and g 2 L2 ( ) We will use these results to conclude (9) has a unique solution and, in the next section, to form an error estimate for this method. Lemma 2: Suppose V 2 Shk satis es (17) then k tVxk2 + k jxj1=2 V k2 1 + k jxj1=2 V k2 0 + Ckgk2 (18) Proof: Let = tV 2 Shk in (17), sum from n = 1; : : : ; N, and observe that Vt 2 Thk to see that (xVt ; V ) + ( t Vx ; t Vx ) = (g; t V ) : 5 From the Poincare inequality (8), arithmetic-geometric mean inequality (6), and the IC/PC on V we have 1 xV 2 ; 1 1 1 xV 2 ; 1 0 + k t Vx k2 4 kgk2 + k t V k2 4 kgk2 + C k t Vx k2 : 1 2 + The inequality (18) now follows from this by choosing = 1=C.2 The next lemma gives a bound on the approximation functions at the time nodes, tn . Lemma 3: Suppose V 2 Shk satis es (17) then there is a constant C such that kxV nk2 2 C kxV k 0 + kxV k 1 + kgk + (1 + h 2 )k t Vx k2 + kxV k2 ; + k 2 2 2 ! (19) where 0 < < 1. Proof: Take = x t (xV ) in (17) and obtain (xVt ; x t (xV ))Sn = (Vx ; ( x t (xV ))x )Sn + (g; x t (xV ))Sn or x2 Vt ; V Sn = ( t Vx; x ( (x t V ))x )Sn + (g; x (x t V ))Sn + (xVt ; (I x)(x t V ))Sn : Using the inverse inequality (10) and the arithmetic geometric mean inequality (6) we obtain 1 n2 1 n12 1 2 1 2 2 kxV k 2 kxV k + 2 kgkSn + 2 k x (x t V )kSn + k t Vx kSn k x (x t V )x kSn +Ck 1 kxV kSn k(I x )(x t V )kSn : Applying (6), the estimate (13), the Poincare inequality (8), (14), and (16) we have 1 kxV n k2 1 kxV n 1 k2 + 1 kgk2 + C(1 + h2 )k V k2 + =2kxV k2 : Sn 2 2 2 Sn k 2 t x Sn Iterating this inequality we obtain The estimate (19) now follows by adding kxV n k 2 kxV nk2 kxV 0k2 + kgk2 + C(1 + h 2 )k t Vxk2 + kxV k2 : k + 1 and noting that V 0 0 on +. 0 2 We now are in a position to derive a crucial estimate on the approximation function V . Theorem 1: Suppose V 2 Shk satis es (17) then there is a constant C such that kxV k 2 C(1 + h2 )kgk : k (20) 6 Proof: From lemma 1 with = 1, the weight function inequality (16), and the Poincare inequality (8) we have kxV k2 k Using (19) we obtain N X n=0 kxV nk2 + Ck tVxk2 : 2 kxV k 2 2 kN C(kxV k 0 + kxV k 1 + kgk + (1 + Ch2 )k t Vx k2 ) + kxV k2 + Ck t Vxk2 ; + k 2 2 " # where = C . Noting that kN = 1, choosing = 1=2, observing that jxj jxj1=2 on 1; 1], and once again using (16) as well as the result of lemma 2 we obtain (20). 2 This estimate allows us to conclude the approximation problem (9) has a unique solution. As noted before (9) leads to a square system of linear equations. By considering the possibility of two solutions we are lead to evaluate the problem with f = 0. From (20) we see that the only solution of the homogeneous problem is the zero solution. Thus, we conclude (9) has a unique solution. In this section we present the main result of this paper. We will use a decomposition of the error involving projection operators similar to standard ones from T]. The nal estimate will involve the norm 2 1 Xd Xd N(u)2 = k( dt )` uk2 2 (I;H p+1 ( )) + k( dx )` uk2 q+1 (I;L2 ( )) L H and the rate quantity `=0 `=0 4. Main Results: (h; k) = (h2p+2 + k2q+2 )1=2 N(u): Theorem 2: Suppose u is the solution of (1){(3) and is su ciently smooth so N(u) < 1. Also suppose U is the solution of (9) then kx(u U)k that u U = (u PxPt u) + (Px Pt u U) = + : From the inequality (16) and the approximation properties of the Px and Pt operators it follows Proof: Let 2 C(1 + h2 ) (h; k) k (21) kx k ku Pxuk + kPx (I Pt )uk 7 C (h; k): (22) Note that 2 Shk . For 2 Thk we have (x t ; )Sn + ( x; x )Sn = = = = where (x(Pt Px u)t ; )Sn + ((Pt Px u)x ; x )Sn (f; )Sn (xPx ut ; )Sn + (Pt ux ; x )Sn (xut ; )Sn (ux ; x )Sn (x(Px I)ut ; )Sn + ((Pt I)u)x ; x )Sn (R; )Sn Combining this with (22) proves the theorem.2 R = x(Px I)ut (Pt I)uxx: From (20) and the approximation properties of Px and Pt we have 2 2 kx k C(1 + h2 )kRk C(1 + h2 ) (h; k): k k In this section we describe the results of two computational experiments with the cG nite element scheme set in the simplest case when p = q = 1 and h = k. The examples are designed so that the true solution is known. This allows us to verify that the weighted error, kx(u U)k , will tend to zero at rate O(h2 ). We rst brie y describe our implementation. Since the test functions in the q = 1 case are constant in t on each slab, Sn , it can be shown that (9) in the p = q = 1 is equivalent to Z tn+1 f( ; t) dt; (23) k 1 x(U n+1 U n ); + 1 (U n + U n+1 )x ; x = k 1 2 tn 1 for n = 0; 1; : : : ; N 1 and 2 Xh . We used Simpson's rule to compute the inner product and time integrations that occur on the right side of the equation. Since this rule is very accurate at O(h4 ) we do not expect the approximation of this term will alter the convergence rate of the overall scheme. There are many possible approaches to solving the equations that arise in numerical approximation schemes for forward-backward parabolic problems (see VK] for an iterative scheme that solves the forward and backward parts separately and AFJK] for a full list of possible schemes). Since our focus is on verifying the convergence theorem we take the test functions to be the elements in 1 the usual hat function basis for Xh and obtain a system equations for the unknown values of U at the nodes, Ujn = U(xj ; tn ). We solve the resulting system using Gaussian Elimination. In the succeeding two experiments we chose a function for u, substituted it into (1), and obtained a right side function f. The integrations in the computations of the L2 ( )-norm of the errors are 5. Numerical Results: 8 approximated using the Trapezoid rule in each example. The solutions in each case are smooth guarranteeing that N(u) < 1. In example 1 the function u satis es the BC/IC/PC and thus the theorem applies with the exception of the Simpson's rule approximation on the right side term. In example 2 the theorem does not apply since the known solution does not satisfy the BC/IC/PC. However in both cases we see that the ratio of the error to h2 is tending to a constant con rming the theorem. Example 1: u(x; y) = cos( 1=2 1=4 1=8 1=16 h x=2)sin( x). kx(u 5.9074(-2) 1.3324(-2) 3.2163(-3) 7.9725(-4) U)k kx(u 0.236 0.213 0.206 0.204 U)k =h2 Example 2: u(x; y) = cos(x)cos(x). 1=2 1=4 1=8 1=16 h kx(u 3.6568(-3) 7.7141(-4) 1.8442(-4) 4.5577(-5) U)k kx(u 0.0146 0.0123 0.0118 0.0117 U)k =h2 6. Conclusions: In this paper we have described a space-time nite element method for a forward-backward parabolic problem and proved an optimal a priori error estimate in a weighted norm. We have also performed some computations that display this result. We now compare our convergence result to others in the literature. The equation we study has the form ut uxx = f. We have restricted our analysis to the case where (x; t) = x in order to keep the arguments straightforward. Here, we have shown O(hp+1 ) convergence in a weighted L2 -norm. In VK] a rst-order in time, second-order in space, nite di erence method is analyzed and the L1-norm of the error at the nodes is shown to converge at rate O(h2 + k) under the restriction that = (x). In AL1], AL2], and AFJK] the function is allowed to depend on x and t. A rate of O(hp ) was shown for the L2 ( )-norm of the error in AL1] and AL2]. This O(hp ) rate was shown for the L2 ( )-norm of the x-derivatives of the error in AL2] and AFJK] which is optimal. We showed O(hp ) convergence for the L2 -norm of x-derivative of the error in F] where the dG nite element method is considered with (x; t) = x. 9 We anticipate that some generalizations of our theorem are possible. We expect that if is independent of t and smooth the argments above can be generalized with di erent weight functions. However we expect the dependence of on t must be restricted so the curves separating regions where > 0 from those where < 0 are parallel or close to parallel the t-axis. In contrast, we expect that the dG scheme (see F]) will generalize mainly because one does not have to count nodes as carefully as was needed for cG (see section 2). References AFJK] A.K. Aziz, D.A. French, S. Jensen, and R.B. Kellogg, Origins, analysis, and numerical analysis of a forward-backward parabolic problem (submitted to M 2 AN). AL1] A.K. Aziz and J.-L. Liu, A Galerkin method for the forward-backward heat equation, Math. Comp. 56 (1991), 35-44. AL2] A.K. Aziz and J.-L. Liu, A weighted least squares method for the backward-forward heat equation, SIAM J. Num. Anal. 28 (1991), 156-167. AM] A.K. Aziz and P. Monk, Continuous nite elements in space and time for the heat equation, Math. Comp. 52 (1989), 255-274. BDK] J.L. Bona, V.A. Dougalis, and O.A. Karakashian, Fully discrete Galerkin methods for the Kortewig-de-Vries equation, Comput. Math. Appl. 12A (1986), 859-884. BG] M.S. Baouendi and P. Grisvard, Sur une equation d'evolution changeant de type, J. Funct. Anal. 28 (1968), 352-367. B] R. Beals, An abstract treatment of some forward backward problems in transport and scattering, J. Funct. Anal. 34 (1979), 1-20. C] P.G. Ciarlet, The nite element method for elliptic problems, North Holland, Amsterdam (1978). EF] D. Estep and D.A. French, Global error control for the continuous Galerkin nite element method for ordinary di erential equations, RAIRO 28 (1994), 815-852. FR] J.N. Franklin and E.R. Rodemich, Numerical analysis of an elliptic-parabolic partial di erential equation, SIAM J. Num. Anal 5 (1968), 680-716. F] D.A. French, Discontinuous Galerkin nite element methods for a forward{backward heat equation, Appl. Numer. Math. 27 (1998), 1-8. 10 FJ] D.A. French and S. Jensen, Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems, IMA J. Num. Anal. 14 (1994), 421-442. FP] D.A. French and T.E. Peterson, A continuous space-time nite element method for the wave equation, Math. Comp., 65 (1996), 491-506. FS] D.A. French and J.W. Schae er, Continuous nite element methods which preserve energy properties for nonlinear problems, Appl. Math. Comp. 39 (1990), 271-295. GM] J.A. Goldstein and T. Mazumdar, A heat equation in which the di usion coe cient changes sign, J. Math. Anal. Appl. 103 (1984), 533-564. T] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics # 1054, Springer Verlag, Berlin (1984). VK] V. Vanaja and R.B. Kellogg, Iterative methods for a forward-backward heat equation, SIAM J. Num. Anal. 27 (1990), 622-635. 11

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