When 12 we have to satisfy the condition 1 pt 2 h 61

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Unformatted text preview: tually improves slightly to p t=h2 1=2. Let us now turn to a more general examination of stability and convergence. Let's again focus on our model problem: determine u 2 H01 satisfying If (v ut) + A(v u) = (v f ) (v u) = (v u0) 8v 2 H01 8v 2 H01 t>0 t = 0: (9.4.9b) N The semi-discrete approximation consists of determining U 2 S0 (V Ut ) + A(V U ) = (V f ) 8V 2 S0N (9.4.9a) H01 such that t>0 (9.4.10a) 9.4. Convergence and Stability 23 (V U ) = (V u0) 8V 2 S0N t = 0: (9.4.10b) Trivial Dirichlet boundary data, again, simpli es the analysis. Our rst result establishes the absolute stability of the nite element solution of the semi-discrete problem (9.4.10) in the L2 norm. Theorem 9.4.1. Let 2 S0N satisfy 8V 2 S0N (V t ) + A(V ) = 0 (V ) = (V 0 ) 8V 2 S0N t>0 t = 0: (9.4.11a) (9.4.11b) Then k ( t)k0 k 0k0 t > 0: (9.4.11c) Remark 1. With (x t) being the di erence between two solutions of (9.4.10a) satisfying initial conditions that di er by 0 (x), the loading (V f ) vanishes upon subtraction (as with (9.4.2)). Proof. Replace V in (9.4.11a) by to obtain ( or Integrating t ) + A( ) = 0 1 d k k2 + A( ) = 0: 2 dt 0 k ( t)k = k ( 0)k ; 2 2 0 2 0 Zt 0 A( )d : The result (9.4.11c) follows by using the initial data (9.4.11b) and the non-negativity of A( ). We've discussed stability at some length, so now let us turn to the concept of convergence. Convergence analyses for semi-discrete Galerkin approximations parallels the lines of those for elliptic systems. Let us, as an example, establish convergence for piecewiselinear solutions of (9.4.10) to solutions of (9.4.9). Theorem 9.4.2. Let S0N consist of continuous piecewise-linear polynomials on a family of uniform meshes h characterized by their maximum element size h. Then there exists a constant C > 0 such that max ku ; U k0 t2(0 T ] T C (1 + j log h2 j)h2 tmax] kuk2: 2(0 T (9.4.12) 24 Parabolic Problems N Proof. Create the auxiliary problem: determine W 2 S0 such that ;(V W ( )) + A(V W ( 8V 2 S0N )) = 0 2 (0 t) ~ W (x y t) = E (x y t) = U (x y t) ; U (x y t) (9.4.13a) (9.4.13b) ~N where U 2 S0 satis es A(V u( ~ ) ; U( )) = 0 8V 2 S0N We see that W satis es a terminal value problem on 0 elliptic problem with as a parameter. Consider the identity 2 (0 T ]: (9.4.13c) ~ t ant that U satis es an d (W E ) = (W E ) + (W E ): d Integrate and use (9.4.13b) kE ( t)k = (W E ( 0)) + 2 0 Use (9.4.13a) with V replaced by E kE ( t)k = (W E ( 0)) + 2 0 Zt 0 Zt 0 (W E ) + (W E )]d : A(W E ) + (W E )]d : (9.4.14) Setting v in (9.4.9) and V in (9.4.10) to W and subtracting yields (W u ; U ) + A(W u ; U ) = 0 >0 (W u ; U )(0) = 0 = 0: Add these results to (9.4.14) and use (9.4.13b) to obtain kE ( t)k = (W ( 0)) + 2 0 Zt 0 A(W ) + (W )]d where ~ = u ; U: The rst term in the integrand vanishes by virtue of (9.4.13c). The second term is integrated by parts to obtain Zt kE ( t)k = (W ( t)) ; (W )d : 2 0 0 (9.4.15a) 9.4. Convergence and Stability 25 This result can be simpli ed slightly by use of Cauchy's inequality (j(W V )...
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