1. Linear System function [A,b,c] = assem_boundary(A,b,p,t,e); np = size(p,2); % total number of nodes bound = unique([e(1,:) e(2,:)]); % boundary nodes inter = setdiff([1:np],bound); % interior nodes b = b(inter); % modify load vector A = A(inter,inter); % modify stiffness matrix c = zeros(np,1); % allocate solution vector c(bound) = sparse(length(bound),1); % boundary nodes solution c(inter) = A\b; % solve for free node values function [u1]=HeatSolver2D(p,e,t) k = 0.001; % time step T = 0.07; % final time n = T/k; np = size(p,2); % number of nodes x = p(1,:)'; y = p(2,:)'; % node coordinates u1 = sparse(np,n); % set zero IC ix=find(sqrt(p(1,:).^2+p(2,:).^2)<0.4); u1(ix,1)=ones(size(ix)); M = MassMatrix(p,t); %primary mass matrix A = StiffMatrix(p,t); %primary stiff matrix [A,M] = boundary(A,M,p,t,e); % M,A after imbosing boundary bound = unique([e(1,:) e(2,:)]); % boundary nodes inter = setdiff([1:np],bound); % interior nodes bold = Vec_b(0,p,e,t,bound,inter); for l = 2:n % time loop time = l*k; bnew = Vec_b(time,p,e,t,bound,inter); LHS = (M + 0.5*k*A); % Crank-Nicholson rhs = (M - 0.5*k*A)*u1(inter,l-1) + 0.5*k*(bnew + bold); u1(inter,l) = LHS\rhs; end solve expensive

2. This code simulate two horizontal slices of Model 2 from the 10th SPE Comparative Solution Project. which is publicly available on the net. The model dimensions are 1200 × 2200 × 170 (ft) and the reservoir is described by a heterogeneous distribution over a regular Cartesian grid with 60×220×85 grid- blocks. We pick the top layer, in which the permeability is smooth, and the bottom layer. For both layers, the permeabilities range over at least six orders of magnitude. To drive a flow in the two layers, we impose an injection and a production well in the lower-left and upper-right corners, respectively. 10% 90% MFEM MATLAB code applied to the top layer of the SPE-10 test case SPE-10SPE-10 with an injection and a production well in the lower-left and upper-right corners, respectively. P I

3. tic; factorial(21); toc

4. Linear System We want to solve the following linear system These methods generate a sequence of approximate solutions Gaussian Elimination LU, Choleski 2 classes of methods Direct Methods Iterative Methods

5. observations made by Horst Simon [173]. He predicted that by now we will have to solve routinely linear problems with some 5 × 10^9 unknowns. From extrapolation of the CPU times observed for a characteristic model problem, he estimated the CPU time for the most efficient direct method as 520 040 years, provided that the computation can be carried out at a speed of 1 TFLOPS. On the other hand, the extrapolated guess for the CPU time with preconditioned conjugate gradients, still assuming a processing speed of 1 TFLOPS, is 575 seconds. ‘flops’ floating point operations. Each addition, subtraction, multiplication, division, or square root counts as one flop.

6. Linear System We want to solve the following linear system 2 classes of methods Direct Methods Iterative Methods Krylov subspace splitting others Jacobi, GS, SOR,..PCG, MINRES, GMRES, …

7. Linear System Remark: (1) has a unique solution We want to solve the following linear system A is invertable Remark: A is invertable det(A)=0 Remark: A is invertable Example: Solve: 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8

8. Linear System We want to solve the following linear system Remark: Definition: Example:

9. Iterative Method These methods generate a sequence of approximate solutions Remark:

10. DEF: with eigenvalues Splittings and Convergence DEF: spectral radius of A is defined to be Splitting A large family of iteration

11. Splittings and Convergence A large family of iteration DiagonalLowerUpper Jacobi:Gauss-Seidel:

12. Jacobi Method Consider 4x4 case 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 Example

13. K=6K=7K=8K=9K=10 x1 x2 x3 X4 K=1K=2K=3K=4K=5 x1 x2 x3 X4 Jacobi Method 0.6000 1.0473 0.9326 1.0152 0.9890 2.2727 1.7159 2.0533 1.9537 2.0114 -1.1000 -0.8052 -1.0493 -0.9681 -1.0103 1.8750 0.8852 1.1309 0.9738 1.0214 1.0032 0.9981 1.0006 0.9997 1.0001 1.9922 2.0023 1.9987 2.0004 1.9998 -0.9945 -1.0020 -0.9990 -1.0004 -0.9998 0.9944 1.0036 0.9989 1.0006 0.9998 11.3537 4.9910 2.0299 0.8911 0.3686 0.1605 0.0671 0.0290 0.0122 0.0053

14. K=1K=2K=3K=4K=5 x1 x2 x3 X4 Gauss Seidel Method 0.6000 1.0302 1.0066 1.0009 1.0001 2.3273 2.0369 2.0036 2.0003 2.0000 -0.9873 -1.0145 -1.0025 -1.0003 -1.0000 0.8789 0.9843 0.9984 0.9998 1.0000 5.6930 0.4300 0.0662 0.0082 0.0009 Note that in the Jacobi iteration one does not use the most recently available information.

15. Gauss-Seidel iteration for general n: Gauss Seidel Method Jacobi iteration for general n:

16. Splittings and Convergence THM: A large family of iteration

17. Splittings and Convergence Example: 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 10 11 10 8 2 -1 0 3 -1 -1 2 0 -1 3 Jacobi: U=triu(A,1) L=tril(A,-1) D=diag(diag(A)) eig(inv(M)*N) -0.4264 -0.1040 0.1860 0.3445 GS: 0 0 0.0839 - 0.0322i 0.0839 + 0.0322i

18. Splittings and Convergence THM: A large family of iteration Proof: (Golub p511) Remarks:

19. Splittings and Convergence THM: Proof: (Golub p512) show that all eigenvalues are less than one. Positive Definite Symmetric Stiffness Matrix Mass Matrix

20. Example: Splittings and Convergence DEF: IF THM:

21. Successive over Relaxation The Gauss-Seidel iteration is very attractive because of its simplicity. Unfortunately, if the spectral radius is close to one, then convergence is vey slow. One solution for this Successive over Relaxation GS:

22. Successive over Relaxation Example: 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8

23. K=1K=2K=3 x1 x2 x3 X4 Successive over Relaxation Example: 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 0.6180 1.0553 1.0055 2.3988 2.0273 2.0004 -1.0132 -1.0211 -1.0014 0.8743 0.9905 1.0000

24. MATLAB CODE Jacobi iteration for general n: Ex: Write a Matlab function for Jacobi function [sol,X]=jacobi(A,b,x0) n=length(b); maxiter=10; x=x0; for k=1:maxiter for i=1:n sum1=0; for j=1:i-1 sum1=sum1+A(i,j)*x(j); end sum2=0; for j=i+1:n sum2=sum2+A(i,j)*x(j); end xnew(i)=(b(i)-sum1-sum2)/A(i,i) end X(1:n,k)=xnew; x=xnew; end sol=xnew;

25. MATLAB CODE GS iteration for general n: Ex: Write a Matlab function for GS function [sol,X]=gs(A,b,x0) n=length(b); maxiter=10; x=x0; for k=1:maxiter for i=1:n sum1=0; for j=1:i-1 sum1=sum1+A(i,j)*x(j); end sum2=0; for j=i+1:n sum2=sum2+A(i,j)*x(j); end x(i)=(b(i)-sum1-sum2)/A(i,i) end X(1:n,k)=x; end sol=x;

26. Another Look Remark: Given: We want to improve this approximate: Jacobi: GS:

27. Examples of Splittings 1) Non-symmetric Matrix: 2) Domain Decomposition: symmetricSkew-symmetric