Weak Formulation BVP Weak Formulation variational formulation Multiply

Weak Formulation BVP Weak Formulation ( variational formulation) Multiply equation (1) by and then integrate over the domain Green’s theorem gives where

Green’s First identity in R^2 (p 285) Green’s First Identity

Galerkin Methods Weak Formulation ( variational formulation) Infinite dimensional space Is finite dim Unique sol?

Galerkin Methods Discrete Form Is finite dim We can approximate u

Galerkin Methods Linear system Unique sol? 1) 2) 3) 4) Linear system of equation square Symmetric (why) Positive definite

Finite Element Methods No of elements = 16 No of nodes = 13 No interior nodes = 5 No of boundary nodes = 8 Triangulation why

Finite Element Methods 1 0. 25 0. 75 1 0 0. 25 0. 75 1 0 1 D Problem 1 1

Global basis functions

Element Labeling 6 14 3 11 15 2 5 10 9 13 12 7 4 16 1 8

Node Labeling 2 (global labeling) 6 11 7 10 5 12 3 1 9 13 8 4

global basis functions 2 6 11 7 10 5 12 3 1 9 13 8 4

Global basis functions

global basis functions 2 6 0 0 11 0 12 13 0 8 0 0 5 0 3 0 10 0 7 0 1 9 0 0 4

6 14 2 3 5 10 15 11 9 13 12 7 1 4 16 8

global basis functions 2 6 1 0 0 11 10 0 7 0 0 9 0 12 13 0 3 5 0 8 0 0 4

Assemble linear system 2 6 11 7 10 5 12 3 1 9 13 8 4 6 14 2 3 5 10 15 11 9 13 12 7 1 4 16 8

Assemble linear system 2 6 0 7 10 5 0 0 13 0 0 2 0 0 0 8 4 6 0 7 0 11 0 0 0 12 0 9 0 12 3 3 0 11 0 1 10 5 0 0 8 9 13 0 0 4

Finite Element Methods Home. Work: Compute the matrix A and the vector b then solve the linear system and write the solution as a linear combination of the basis then approximate the value of the function at (x, y)= (0. 3, 0. 3) and (0. 7, 0. 7). can you find the analytic solution of the problem? where f(x)= x(x-1)y(y-1) with pcw-linear

Approximation of u 1 0 2 0 3 0 4 0 5 0. 069 6 0 7 0 8 0 9 0 10 0. 049 11 0. 049 12 0. 049 13 0. 049

Matlab matrices (computation info) X-coordinate and y-coordinate Matrix p(2, #elements) 1 2 3 4 5 6 7 8 9 10 11 12 13 x 1 0 0 1 0. 5 0 0. 5 1 0. 75 0. 25 0. 75 y 1 1 0 0 0. 5 1 0. 5 0. 75 0. 25

Matlab matrices (computation info) Boundary node vector e(#boundary node) e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 start 1 2 3 4 6 7 8 9 end 6 7 8 9 2 3 4 1

Matlab matrices (computation info) Node Label (local labeling) Each triangle has 3 nodes. Label them locally inside the triangle 3 1 2

Matlab matrices (computation info) Node and Element Label

Matlab matrices (computation info) Local label. vs. global label Matrix t(3, #elements) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 4 1 2 3 4 5 5 10 11 12 13 2 9 6 7 8 10 11 12 13 13 10 11 12 9 6 7 8 10 11 12 13 9 6 7 8