3 1 Laplaces equation Christopher Crawford PHY 311

§ 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 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 – HW 5 2

Summary of Ch. 2 3

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

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

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: 6

Classification of – Same as conic sections (where nd 2 order PDEs ) • 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 – Timelike (initial) and spacelike (edges) boundaries – 1 initial condition in time, 1 boundary condition at each edge 7

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

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: 9

Solution: relaxation method 1. Discretize Laplacian 2. Fix boundary values 3. Iterate adjusting potentials on the grid until solution settles 10

Solution: finite difference method 1. Discretize Laplacian 2. Fix boundary values 3. Solve matrix equation for potential on grid 11

Solution: finite element method 1. Weak formulation: integral equation 2. Approximate potential by basis functions on a mesh 3. Integrate basis functions; solve matrix equation 12