Classical Electrodynamics · Problem 2.30

Boundary-Value Problems in Electrostatics: II — John David Jackson

Physics — E&M · galerkin-method-poisson-equation-lattice

Problem

Using the results of Problem 2.29, apply the Galerkin method to the integral equivalent of the Poisson equation with zero potential on the boundary, ∫Vdx dy[∇ϕi,j⋅∇ψ−4πρϕi,j]=0withψ(x,y)=∑i′,j′=1Nψi′,j′ϕi′,j′(x,y)\int_V dx\,dy[\mathbf{\nabla}\phi_{i,j}\cdot\mathbf{\nabla}\psi - 4\pi\rho\phi_{i,j}] = 0 \quad \text{with} \quad \psi(x, y) = \sum_{i',j'=1}^{N}\psi_{i',j'}\phi_{i',j'}(x, y) for the lattice of Problem 1.24, with its three independent lattice sites. Show that you get three coupled equations for the ψi,j\psi_{i,j} values (ψ1,ψ2,ψ3\psi_1, \psi_2, \psi_3) and solve to find the ``Galerkin'' approximations for the potential at these sites. Compare with the exact values and the results of the various iterations of Problem 1.24c. Comment. [ψ=4πϵ0Φ][\psi = 4\pi\epsilon_0 \Phi].

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