Introduction to Piecewise Polynomial Spaces and Continuous Linear Functions

1. Piecewise Polynomial Spaces The reason for introducing a mesh of a domain is that it allows for a construction of piecewise polynomial function spaces on this domain Definition: linear function in x and y Is a linear function in x and y Example:

2. Piecewise Polynomial Spaces Definition: triangle with nodes be the space of linear functions on We observe that any member in is uniquely determined by its nodal values Remark: Find a linear polynomial on the triangle K such that p(N1)=2 , p(N2) = 3, p(N3)=1 Example:

3. Local basis functions Example: Find a linear function on the triangle K such that p(N1)=2 , p(N2) = 3, p(N3)=1 Example: Find a linear function on the triangle K such that p(N1)=1 , p(N2) = 0, p(N3)=0 p(N1)=0 , p(N2) = 1, p(N3)=0 p(N1)=0 , p(N2) = 0, p(N3)=1 Example: Remark: any member in can be expressed as a linear combination of these three functions

4. Local basis functions The local basis functions for the triangle K are

5. Reference triangle Local basis functions Exercise3 Find three linear functions on the reference triangle such that Then find a linear function on the reference triangle K such that p(0,0)=2 , p(1,0) = 3, p(0,1)=1

6. Continuous Piecewise Polynomial Spaces Definition: the space of all continuous functions Definition: be a triangulation of the space of all continuous piecewise linear polynomials An example of a continuous piecewise linear function

7. Global Basis Functions for the space of all continuous piecewise linear polynomials To construct a basis for this space we note that a function v in this space is uniquely determined by its nodal values where n is the number of nodes in the mesh

8. Example (for global basis functions)

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

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

11. Global basis functions related to interior nodes

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

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

14. 2 6 1 0 0 11 10 0 0 7 5 9 0 0 0 12 13 0 0 0 3 8 4 Exercise4: Find in explicit form

15. Continuous Piecewise Linear Interpolation Definition: Let we define its continuous piecewise linear interpolant by Remark: approximates by taking on the same values in the nodes Ni.

16. to draw πf given f [p,e,t] = initmesh('squareg','hmax',0.7); % mesh x = p(1,:); y = p(2,:); % node coordinates pif = x.^2+ y.^2; % nodal values of interpolant pdesurf(p,t,pif') % plot interpolant %pdeplot(p,e,t,'xydata',pif,'zdata',pif,'mesh','on');

17. Reference triangle Piecewise Polynomial Spaces Definition: triangle with nodes be the space of linear functions on Remark: We observe that any function inP1(K) is uniquely determined by its nodal values