1. 1DFEM Implementation Example: % example_1D.m files clear n = 7; % no interior points h = 1/(n+1); a = inline('x+2','x'); c = inline('1','x'); f = inline('-x^2+5*x+3','x'); uexact = inline('x*(1-x)','x'); K = sparse(n,n); M1 = sparse(n,n); L = sparse(n,1); x = h:h:1; i=1;K(i,i)=(a((x(i)+x(i+1))/2)+a(x(i)/2))/h; i=n;K(i,i)=(a((x(i)+x(i+1))/2)+a((x(i)+x(i-1))/2))/h; for i=2:n-1 K(i,i)=(a((x(i)+x(i+1))/2)+a((x(i)+x(i-1))/2))/h; end for i=1:n-1 K(i,i+1) = -a( (x(i)+x(i+1))/2 )/h; K(i+1,i) = K(i,i+1); end v = ones(n, 1) ; M1 = diag(4*v,0)+diag(v(1:n-1),-1)+diag(v(1:n-1),1); M = (h/6)*M1; A = K + M; L(1) = h*(f((x(1))/2)+f((x(1)+x(2))/2))/2; L(n) = h*(f((x(n)+x(n-1))/2)+f((x(n)+1)/2))/2; for i=2:n-1 L(i) = f((x(i)+x(i-1))/2)+f((x(i)+x(i+1))/2); L(i) = h*L(i)/2; end U = A\L uexact(h) Usol = [0;U;0]; xnodes = [0;x']; ezplot('x*(1-x)',[0,1]); hold on plot(xnodes,Usol,'r-o') grid on hold off

2. Local Basis function If we restrict any global basis function on an interval Kj then it becomes either the zero function or one of these local basis functions

3. 1DFEM Implementation Partition of the interval [0,1] Node Coordinate vector Global labeling (nodes) 1 2 3 4 5 6 7 8 9 Element labeling (interval) 1 2 3 4 5 6 7 8 Local labeling (nodes) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2468 1357 Global-local labeling matrix 1 st elem 2ed elem 3ed elem 8 th elem Global label of the first node Global label of the second node Example: Find the coordinate of the 2ed node in the fifth element

4. Three Matrices Partition of the interval [0,1] Nodes Coordinate vector Global local labeling Matrix Boundary nodes vector Mass Matrix stiffness Matrix

5. Assemble the Mass Matrix Mass Matrix n=9 # nodes M=8 # elements Contribution from K 1 Contribution from K 2 Contribution from K 3 Contribution from K 5 Contribution from K 6 Each entries of the matrix is a sum of several Contribution from elements. These contributions are integrals of two local basis functions in the same element

6. Element labeling (interval) 1 2 3 4 5 6 7 8 We generate all possible contributions in each element (this is called local mass matrix) How many integrations ?? 32  24 Assemble the Mass Matrix

7. Local Basis function

8. 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 1 2 1 2 3 4747 3636 4 2121 3232 2 3 4 5757 4646 5

9. 1DFEM Implementation % main.m file p=[0,0.2,0.4,0.45,0.5,0.55,0.6,0.8,1]; t=[1:8; 2:9]; e=[1,9]; [M] = createM(p,e,t) % createM.m file function [M] = createM(p,e,t) n = length(p); % number of nodes m = size(t,2); % number of elements M = sparse(n,n); for j=1:m loc2glb = t(1:2,j); % global labeling for the nodes in Kj xcoord = p(loc2glb); % coordinates of the nodes hj = xcoord(2)-xcoord(1); l_mass = (hj/6)*[2, 1;... 1, 2]; % local mass matrix M(loc2glb,loc2glb) = M(loc2glb,loc2glb) + l_mass; % Add contributions end

10. Stiffness Matrix Stiff Matrix

11. % main.m file p=[0,0.2,0.4,0.45,0.5,0.55,0.6,0.8,1]; t=[1:8; 2:9]; e=[1,9]; [K] = createK(p,e,t) Stiffness Matrix % createK.m file function [K] = createK(p,e,t) n = length(p); % number of nodes m = size(t,2); % number of elements a = inline('x+2','x'); K = sparse(n,n); for j=1:m loc2glb = t(1:2,j); % global labeling for the nodes in Kj xcoord = p(loc2glb); % coordinates of the nodes hj = xcoord(2)-xcoord(1); xb = (xcoord(1)+xcoord(2))/2; ab = a(xb); l_stif = (ab/hj)*[1, -1;... -1, 1]; % local stiffness matrix K(loc2glb,loc2glb) = K(loc2glb,loc2glb) + l_stif; % Add contributions end

12. Load vector Stiff Matrix Local Load vector

13. % createL.m file function [L] = createL(p,e,t) n = length(p); % number of nodes m = size(t,2); % number of elements f = inline('-x^2+5*x+3','x'); L = sparse(n,1); for j=1:m loc2glb = t(1:2,j); % global labeling for the nodes in Kj xcoord = p(loc2glb); % coordinates of the nodes hj = xcoord(2)-xcoord(1); xb = (xcoord(1)+xcoord(2))/2; fb = f(xb); l_L = (hj/2)*fb*[1; 1]; % local load vector L(loc2glb) = L(loc2glb) + l_L; % Add contributions end % main.m file p=[0,0.2,0.4,0.45,0.5,0.55,0.6,0.8,1]; t=[1:8; 2:9]; e=[1,9]; [L] = createL(p,e,t) Load vector

14. Weak Boundary Conditions Weak Example: Consider the partition 1 2 3 4

15. Boundary Conditions Mass Matrix stiffness Matrix impose boundary conditions 1 2 3 4 5 6 7 8 9 We need to remove contributions coming from these two red local functions for i=1:length(e) ib = e(i); A(ib,:)=sparse(1,n); A(:,ib)=sparse(n,1); A(ib,ib)=1; L(ib)=0; end

16. Boundary Conditions % non uniform mesh p=[0,0.2,0.4,0.45,0.5,0.55,0.6,0.8,1]; t=[1:8; 2:9]; e=[1,9]; % % uniform mesh % p=0:0.125:1; n = length(p); t=[1:n-1;2:n]; % e=[1,n]; n = length(p); % number of nodes m = size(t,2); % number of elements uexact = inline('x*(1-x)','x'); [M] = createM(p,e,t) [K] = createK(p,e,t) [L] = createL(p,e,t) A = K + M; for i=1:length(e) ib = e(i); A(ib,:)=sparse(1,n); A(:,ib)=sparse(n,1); A(ib,ib)=1; L(ib)=0; end U = A\L; ezplot('x*(1-x)',[0,1]); hold on; plot(p,U,'r-o'); grid on; hold off Impose boundary conditions

17. Uniform Mesh Uniform mesh (using 8 elements) Non-uniform mesh (using 8 elements)

18. Higher Order Polynomial a more general finite element space, consisting of piecewise polynomials of degree r − 1, where r is an integer ≥ 2, with the above piecewise linear case included for r = 2, thus Π k denotes the space of polynomials of degree ≤ k. Example: r = 2  S h space of cont pw-linear Example: (HW) r = 3  S h space of cont pw-quadratic 1 2 3 45

19. Example: r = 4  S h space of cont pw-cubic Local mass matrix Local stiffness matrix Higher Order Polynomial

20. 1 2 3 1 2 3 45 6 7 8 9 1 2 1 2 10 34 3 4 4 5 4 5 67 6 7 7 8 7 8 9 9393 Remark: The matrix is not tridiagonal

21. Theorem Approximation properties of S h Theorem 5.1+5.2:

22. Reference element mapping

23. Example: r = 4  S h space of cont pw-cubic Local mass matrix Local stiffness matrix C element 1

24. Example: C element.vs. C element 1 1 C 0 C All global basis functions are continuous (only) All global basis functions and their first derivatives are continuous

25. Weak 4 th order differential equation Weak 1 C 0 C It is required to use element