# 2 semi discrete galerkin problems 3 as usual we apply

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: kin Problems 3 As usual, we apply the divergence theorem to the second-derivative terms in L to reduce the continuity requirements on u. When L has the form of (9.2.1b), the Galerkin problem 1 consists of determining u 2 HE (t > 0) such that (v ut) + A(v u) = (v f )+ < v ; u > 8v 2 H01 t > 0: (9.2.2a) The L2 inner product, strain energy, and boundary inner product are, as with elliptic problems, (v f ) = A(v u) = ZZ ZZ vfdxdy (9.2.2b) p(vxux + vy uy ) + vqu]dxdy (9.2.2c) and < v pun >= Z @N vpunds: (9.2.2d) The natural boundary condition (9.2.1e) has been used to replace pun in the boundary inner product. Except for the presence of the (v ut) term, the formulation appears to the same as for an elliptic problem. Initial conditions for (9.2.2a) are usually determined by projection of the initial data (9.2.1c) either in L2 (v u) = (v u0) 8v 2 H01 t=0 (9.2.3a) or in strain energy A(v u) = A(v u0) 8v 2 H01 t = 0: (9.2.3b) Example 9.2.1. We analyze the one-dimensional heat conduction problem ut = (pux)x + f (x t) 0<x<1 t>0 u(x 0) = u0(x) 0x1 u(0 t) = u(1 t) = 0 t>0 thoroughly in the spirit that we did in Chapter 1 for a two-point boundary value problem. A Galerkin form of this heat-conduction problem consists of determining u 2 H01 satisfying t>0 (v ut) + A(v u) = (v f ) 8v 2 H01 4 Parabolic Problems U(x,t) cj c1 cN-1 x 0 = x0 x1 xj xN-1 xN = 1 Figure 9.2.1: Mesh for the nite element solution of Example 9.2.1. (v u) = (v u0) where A(v u) = Z 8v 2 H01 1 0 t=0 vxpuxdx: Boundary terms of the form (9.2.2d) disappear because v = 0 at x = 0 1 with Dirichlet data. We introduce a mesh on 0 x 1 as shown in Figure 9.2.1 and choose an approxiN mation U of u in a nite-dimensional subspace S0 of H01 having the form U (x t) = X N ;1 j =1 cj (t) j (x): Unlike steady problems, the coe cients cj , j = 1 2 : : : N ;1, depend on t. The Galerkin N nite element problem is to determine U 2 S0 such that ( j Ut ) + A( j U ) = ( j f ) t>0 ( j U ) = ( j u0 ) t=0 j = 1 2 : : : N ; 1: Let us chose a piecewise-linear polynomial basis 8 x;x ; > x ;x ; <; (x) = > xx ;xx k :0 k1 k k1 k+1 k+1 k if xk;1 < x xk if xk < x xk+1 : otherwise This problem is very similar to the one-dimensional elliptic problem considered in Section 1.3, so we'll skip several steps and also construct the discrete equations by vertices rather than by elements. 9.2. Semi-Discrete Galerkin Problems Since j 5 has support on the two elements containing node j we have A( j U ) = Zx j 0 pU j xj ;1 x dx + where ( )0 = d( )=dx. Substituting for j and Ux Z xj 1 cj ; cj;1 Z A( U ) = p(x)( )dx + j xj ;1 hj hj where Zx j +1 0 pU j xj xj +1 x dx ; h 1 p(x)( cj+1 ; cj )dx h j +1 xj j +1 hj = xj ; xj;1: Using the midpoint rule to evaluate the integrals, we have j ; A( j U ) pjh1=2 (cj ; cj;1) ; ph+1=2 (cj+1 ; cj ) j j +1 where pj;1=2 = p(xj;1=2 ). Similarly, ( or ( j Ut ) = Zx j xj ;1 j Ut ) = Zx j xj ;1 U dx + jt c _ j (_j ;1 j ;1 + cj j )dx + Zx j +1 U dx jt xj Zx j +1 j xj (_j j + cj+1 c _ j +1 )dx where (_) = d( )=dt. Since the integrands are quadratic functions of x they may be integrated exactly using...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern