1. Elliptic PDEs and Solvers
LECTURE 3
Elliptic PDEs and Solvers

2. Aim of Lecture This week we will discuss
Overview of Finite Difference Method for Elliptic PDEs
Taylor series
Forward, Backward, Central Differences
Boundary conditions
Using EXCEL to solve PDEs
Jacobi
Gauss-Seidel

3. Taylor Series Consider the function u expanded about the point i,j. ik

4. Finite Difference Equations
Consider the function u expanded about the point x.
Forward Difference
Backward Difference
NOTE: Order of Truncation Error

5. Taylor Series Central difference approximations obtained by:
Similar approximation for y. i
i+1
i-1 j
j +1
j - 1
Central Difference

6. Finite Difference Method
Method based on truncated Taylor series i
i+1
i -1
j -1 j
j+1
ui,j
- Forward difference
- Backward difference
- Central difference
- Central difference
Above approximations can be substituted for differentials in the PDE.

7. Laplace Equation - FDM Consider the Laplace equation. i i+1 i -1 j -1
ui,j
Difference formula for each node:
When h = k we have:

8. Poisson Equation - FDM Consider the Poisson equation. i i+1 i -1 j -1
ui,j
Difference formula for each node:
When h = k we have:

9. Including Boundary Conditions
u = a i
i+1
i -1
j -1 j
j+1
ui,j
Two types of boundary conditions
Dirichlet u = a
Neumann
Boundary conditions must be included into the discretised equations
j -1 j
ui,j
j - 2
i -1 i
i+1

10. Dirichlet Boundary Conditions
ui,j+1 = 2 i
i+1
i -1
j -1 j
j+1
ui,j
Dirichlet Boundary Conditions: e.g. u = 2
Include boundary value directly

11. Example
i =
j =
Consider the following square domain and its representation using a Finite Difference Grid.
Finite difference approximation
u = 1 Y X
(1,0)
(0,0)
(0,1)
(1,1)
u = 0
Need to solve for internal nodal values

12. Example Taking h = k = 1/3 we have: Can show that j = 1 2 3 4

13. Neumann Boundary Conditions
j - 2
j -1 j
ui,j
Either use backward/forward difference
so that
Or use central difference (need to introduce artificial nodes outside the boundary, but note error)
Alternatively, for derivatives in x direction, if
then i
i+1
i -1
j - 2
j -1 j
ui,j

14. Example
i =
j =
Consider the following square domain and its representation using a Finite Difference Grid. Y X
(1,0)
(0,0)
(0,1)
(1,1)
u = 1

15. Example Taking h = k = 1/2 we have: So that
j =
Taking h = k = 1/2 we have:
So that
Dirichlet boundary conditions give
Using central difference approximation
Artificial Node

16. EXCEL FOR SOLVING FINITE DIFFERENCE EQUATIONS

17. USING EXCEL Ideal for solving Finite Difference Equations i i+1 i-1 j

18. Setting Parameters
Write Parameter in Name Box:
Name Box
Formula Box

19. IF command in Excel Using the IF Command
IF(CONDITION, THEN DO THIS, ELSE DO THIS)

20. Setting up FD Grid in EXCEL
Consider Tutorial 3, Question 2
Boundary conditions (grey)
Unknowns (white)
x values (peach)

21. Input FD Formula Formula for D16 Formula for D9 Previous Iteration
Latest Iteration

22. Iteration in Excel
The Function key (F9) updates all values (1 Iteration)
Use the FLAG Parameter
FLAG = 0 (sets all cell values back to 0)
FLAG = 1 (Starts Iterations)
Keep pressing F9 to increment iteration count.
EXCEL updates cells in Columns.
Can program both Jacobi and Gauss-Seidel.

23. Iteration in Excel Consider two iterations (Press F9 twice)
First Iteration
Second Iteration

24. Solution converged to 4 decimal places
Iteration in Excel
Continue Iterating (Pressing F9) until convergence
Solution converged to 4 decimal places

25. Gauss Seidel in Excel
Even easier than Jacobi (don’t require the old values of the previous iteration)
Each cell just uses most up-to-date values
Don’t even know/care which order the cells are calculated in
Consider Tutorial 3, Question 1

26. Using Excel – summary/list of tasks
Set up the parameters of the problem such as FLAG, h, k, g
NB when solving a Laplace equation it is better to set up Excel to solve Poisson and then just set g = 0
Set up the domain (size, x/y values)
Fill in the known (Dirichlet) boundary values
Work out the general FD formula and copy it across all cells with unknown values
Work out FD formulas for all unknown (Neumann) boundary values and copy to relevant cells – next week
There should be no cells inside the domain that reference cells outside the domain (apart from parameters like g or h)
Double click each boundary cell to check