2 03 13 23 33 02 12 22 32 01 11 21 31 00 10

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Unformatted text preview: g01 + g10 ;w11 + 4w21 ; w22 = h2f21 + g31 + g20 ;w11 + 4w12 ; w22 = h2f12 + g02 + g13 ;w21 ; w12 + 4w22 = h2f22 + g32 + g23 fij := f (xi yj ) The algebraic system is 4 4 Order the equations and unknowns by rows 2 32 3 2 2 3 4 ;1 ;1 0 w h f +g +g 6 ;1 4 0 ;1 7 6 w11 7 6 h2f11 + g01 + g10 7 6 6 ;1 0 4 ;1 7 6 w21 7 = 6 h2f21 + g31 + g20 7 76 7 6 4 5 4 12 5 4 12 02 13 7 5 0 ;1 ;1 4 w22 h2f22 + g32 + g23 { The structure of A is not apparent 13 2D Finite Di erences The general nite di erence system { Order the equations and unknowns by rows x = xT xT 1 2 where { Write the di 2 4 ;1 6 ;1 4 6 6 6 6 6 6 6 ;1 6 6 ;1 6 6 6 6 A=6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ;1 ... ;1 ... 4 xT = w1j w2j j 2N ;1 j ] (7a) (7b) erence equation at all interior nodes ;1 ;1 ... 4 ;1 4 ;1 ... ;1 ;1 ;1 ;1 xT ;1]T N ;1 4 ;1 ... ;1 ... ;1 14 4 ;1 ;1 ;1 4 ;1 ... ... ;1 ;1 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2D Finite Di erences The algebraic system is block tridiagonal 2 32 32 3 C1 D1 x1 7 6 b1 7 6 D2 C2 D2 7 6 x2 7 6 b2 7 6 76 6 7 6 .. 7 = 6 .. 7 ... 4 54 54 5 DN ;1 CN ;1 where 2 4 ;1 6 ;1 4 ;1 Cj = 6 6 ... 4 ;1 and 2 f 6 f1j bj =h2 6 ..2j 6 4 fN ; 1 j ; 4 xN ;1 3 7 7 7 5 bN ;1 2 ;1 6 Dj = 6 6 4 ;1 ... (8a) ;1 3 7 7 7 5 (8b) 32 76 7+6 76 6 56 4 2 g 6 g1N 6 2N N ;1 j 6 4 .. gN ;1 N 15 g0j 0 .. 0 gNj 3 7 7 7 5 3 7 7 7; 7 7 5 2 g 6 g10 6 20 1j6 . 4. gN ;1 0 3 7 7 7 5 (8c) Note: 2D Finite Di erences i. Matrix Dimensions: { Cj is a (N ; 1) (N ; 1) tridiagonal matrix { Dj is a (N ; 1) (N ; 1) diagonal matrix { A is a (N ; 1)2 (N ; 1)2 block tridiagonal matrix ii. The Kronecker delta ij is unity when j = k and zero otherwise iii. There are only 5 nonzero bands. The lower and upper bandwidths are p = q = N ; 1 iv. Dj is tridiagonal with nite element discretization { using a piecewise bilinear polynomial basis { The lower and upper bandwidths would be N Fill In for scalar band procedures { uij and lij involve inner products between the ith row of L and the jth column of U { Zero elements within the band of A are nonzero in L and U 16 Fill In Consider N = 4 XX X XXX X XXX X XX X X XX X X XXX X X XXX X X XX X X XX X X XXX X X XXX X X XX X X XX X XXX X XXX X X - Original entry XX - Created by elimination { A has about 5N 2 nonzero entries { L and U require about 2N 3 memory locations { With n N 2 and p = q N Band factorization needs about N 4 multiplications Full factorization needs about N 6=3 multiplications 17 Pro le Schemes { Concentrate on symmetric positive de nite systems { There may be variations in the bandwidth Due to pivoting Due to nonuniform grids { Example 5. Finite element matrix structure Unknowns at vertices are connected to unknowns on elements sharing the vertex Equations and unknowns are ordered as shown 123 2X X 4 3X 3 X 8 9 10 X XXX X XXXX 5 6 67 4X X X X X X X 7 5 5 1X X X X 2 1 4 XXXX 6 10 7 9 8 9 8 10 18 X X X XX XX XX X X XX XX XXX XX XX Pro le Schemes Account for local variation in bandwidth { The skyline or pro le is the envelope formed by the local row or column bandwidths3 4 5 6 7 8 9 10 12 1X X X X 2X X 3X X X XXX X 4X X X X X X X XXXX 5 XX XXXX 6 X 7 X X XX XX X X 9 XX XX 8 XXX XX 10 XX { L and U have the same pro les as A { Let mi be the index of the rst nonzero element of row i lij = 0 and 0 j < mi (9a) k = max(mi mj ) (9b) { The pro le form of the symmetric factorization (3) is d1 = a11 (10a) 1 i;1 X i k=k lji = d (aji ; dk ljk lik ) dj = ajj ; j ;1 X k=mj 19 2 dk ljk i = mj : j ; 1 (10b) j=2:n (10c) Pro le Schemes Th...
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