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

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

Summary of Ch. 2

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

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

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:

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

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

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:

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

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

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