1. Christopher Crawford PHY 311 2014-02-19
§3.1 Laplace’s equation
Christopher Crawford
PHY 311

2. Outline Overview Summary of Ch. 2 Intro to Ch. 3, Ch. 4
Laplacian – curvature (X-ray) operator PDE’s in physics with Laplacian Laplacian in 1-d, 2-d, 3-d
Boundary conditions Classification of hyperbolic, elliptic, parabolic PDE’s External boundaries: uniqueness theorem Internal boundaries: continuity conditions
Numerical solution – real-life problems solved on computer Relaxation method Finite difference Finite element analysis – HW5

3. Summary of Ch. 2

4. Laplacian in physics The source of a conservative flux
Example: electrostatic potential, electric flux, and charge

5. Laplacian in lower dimensions
1-d Laplacian
2nd derivative: curvature
Flux: doesn’t spread out in space
Solution:
Boundary conditions:
Mean field theorem
2-d Laplacian
Flux: spreads out on surface
2nd order elliptic PDE
No trivial integration
Depends on boundary cond.
No local extrema

6. Laplacian in 3-d Laplace equation:
Now curvature in all three dimensions – harder to visualize
All three curvatures must add to zero
Unique solution is determined by fixing V on boundary surface
Mean value theorem:

7. Classification of 2nd order PDEs
Same as conic sections (where )
Elliptic – Laplacian
Spacelike boundary everywhere
1 boundary condition at each point on the boundary surface
Hyperbolic – wave equation
Timelike (initial) and spacelike (edges) boundaries
2 initial conditions in time, 1 boundary condition at each edge
Parabolic – diffusion equation
1 initial condition in time, 1 boundary condition at each edge

8. External boundary conditions
Uniqueness theorem – difference between any two solutions of Poisson’s equation is determined by values on the boundary
External boundary conditions:

9. Internal boundary conditions
Possible singularities (charge, current) on the interface between two materials
Boundary conditions “sew” together solutions on either side of the boundary
External: 1 condition on each side Internal: 2 interconnected conditions
General prescription to derive any boundary condition:

10. Solution: relaxation method
Discretize Laplacian
Fix boundary values
Iterate adjusting potentials on the grid until solution settles

11. Solution: finite difference method
Discretize Laplacian
Fix boundary values
Solve matrix equation for potential on grid

12. Solution: finite element method
Weak formulation: integral equation
Approximate potential by basis functions on a mesh
Integrate basis functions; solve matrix equation