1. Scientific Computing Partial Differential Equations Poisson Equation Calculus of Variations

2. Finite Differences In our last lecture we looked at the oldest, and perhaps the simplest, method for solving the Poisson Equation – the method of Finite Differences using the 2 nd order Centered difference formula. While this method is suitable for a wide range of problems, it suffers from the restriction of the domain being rectangular in shape.

3. Finite Element Method The Finite Element Method was developed to address this problem. This method can handle a variety of complex domain geometries. In this method, we do not approximate derivatives to find a solution. Rather, we change the problem to an associated minimization principle from the Calculus of Variations.

4. Calculus of Optimization Recall from single variable calculus: Continuous functions must take extreme values on a bounded domain. A necessary condition for an extremum at x 0, if f is differentiable, is f’(x 0 )=0

5. Calculus of Optimization For scalar multi-variable functions F(x) = F(x 1,..., x n ) how do we find an extremal value? Consider the function g(t) = F(x + t v) = F(x 1 + t v 1,..., x n + t v n ) where v is some non-zero vector. If x is an extremum for F, then d/dt g(t)| t=0 = 0

6. Calculus of Optimization Using the chain rule, we get: At t=0: This equation must be true for all vectors v. Thus, if F has an extremum at x, then

7. Calculus of Variations In the calculus of variations, we extend this optimization principle from optimizing functions to optimizing functionals. Example: The minimal curve problem is to find the shortest path between two specified locations. In simplest form, we are given two points a = (a 1, a 2 ) and b = (b 1, b 2 ). We want to find the curve of shortest length connecting them. a b

8. Calculus of Variations Let us assume the curve is the graph of a differentiable function y=u(x). Then, the length of the curve is given by the standard arclength integral: We would require that u(x) satisfy the boundary condition s: a 2 =u(a 1 ) and b 2 =u(b 1 ). Then, the task is to find the function u(x) that will minimize the integral, among all possible functions satisfying the boundary conditions. The integral is called a functional.

9. Calculus of Variations The general problem in the calculus of variations is to find a function that optimizes some functional. Note the difference between optimizing a function and optimizing a functional: Function: Find optimum for f(x) among points on an interval. Functional: Find optimum for I[u] among a space of possible functions (often this space is infinite-dimensional) What is the condition for optimizing a functional that is analogous to analogous to for functions?

10. Calculus of Variations Single Variable: Suppose that the functional I[u] is an integral value of x,u,and u x : As we did before, let’s add a term to vary the function u where v(x) is a function satisfying the same boundary conditions as u(x). We get If u(x) optimizes the functional, we must have g’(0)=0.

11. Calculus of Variations Assuming the integral and functions u and v are sufficiently smooth, we can bring the derivative inside the integral to get: where we let p = w x. Then, using w = u+tv, we get If u is an optimum, then g’(0)=0. Also, w| t=0 =u and w x | t=0 =u x. Thus,

12. Calculus of Variations Now, For the second integral, we use integration by parts: To insure that u and w satisfy the same boundary conditions, we must have v(a)=v(b)=0. Thus, we have So, for all functions v(x).

13. Calculus of Variations Since this formula holds for all variation functions v(x), then it must be the case that the integrand is identically zero. That is, This result is known as the Euler-Lagrange Equation

14. Calculus of Variations Multi-Variable: Consider the task of finding the optimum function u(x,y) for an integral functional As with the single variable case, we create a variational function w(x,y) = u(x,y)+tv(x,y) where v is zero on the boundary of D.

15. Calculus of Variations Again, by using integration by parts on the second and third terms, we get for all variation functions v and so,

16. Calculus of Variations With this method, the E-L equation can be extended to n variables: Note: – Process gives an extremum: distinguishing mathematically between max/min is difficult. We usually have to use geometry of physical setup to tell if we have a max or min. – Solution function u must have continuous second-order derivatives - requirement from integration by parts

17. Calculus of Variations Example: Curve of shortest length had functional E-L : Here, the integrand is Now, and

18. Calculus of Variations Example: So, The only solution of this is u’’=0, that is a linear function in x, which is not surprising!

19. The Poisson Equation This equation can be generated as the solution to a Calculus of Variations problem. Let The E-L equation is

20. The Poisson Equation So, Thus, we get the E-L equations lead to the Poisson Equation.