§3.1 Laplace’s equation

Christopher CrawfordPHY 311

2014-02-19

Outline• Overview

Summary of Ch. 2Intro to Ch. 3, Ch. 4

• Laplacian – curvature (X-ray) operatorPDE’s in physics with LaplacianLaplacian in 1-d, 2-d, 3-d

• Boundary conditionsClassification of hyperbolic, elliptic, parabolic PDE’sExternal boundaries: uniqueness theorem Internal boundaries: continuity conditions

• Numerical solution – real-life problems solved on computerRelaxation methodFinite differenceFinite element analysis – HW5

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Summary of Ch. 2

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Laplacian in physics• The source of a conservative flux

– Example: electrostatic potential, electric flux, and charge

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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– 2nd derivative: curvature– Flux: spreads out on surface– 2nd order elliptic PDE– No trivial integration

• Depends on boundary cond.

– Mean field theorem• No local extrema

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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:

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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– Timelike (initial) and spacelike (edges) boundaries– 1 initial condition in time, 1 boundary condition at each edge

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External boundary conditions• Uniqueness theorem – difference between any two solutions of

Poisson’s equation is determined by values on the boundary

• External boundary conditions:

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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:

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Solution: relaxation method1. Discretize Laplacian2. Fix boundary values3. Iterate adjusting potentials on the grid until solution settles

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Solution: finite difference method1. Discretize Laplacian2. Fix boundary values3. Solve matrix equation for potential on grid

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Solution: finite element method1. Weak formulation: integral equation2. Approximate potential by basis functions on a mesh3. Integrate basis functions; solve matrix equation

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