Classical Electrodynamics · Problem 2.29

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

Physics — E&M · galerkin-method-square-lattice-integrals

Problem

For the Galerkin method on a two-dimensional square lattice with lattice spacing hh, verify the relations (2.81) for the localized ``pyramid'' basis functions, ϕij(x,y)=(1−∣x∣/h)(1−∣y∣/h)\phi_{ij}(x, y) = (1 - |x|/h)(1 - |y|/h), ∣x∣<h|x| < h, ∣y∣<h|y| < h, where xx and yy are measured from the site (i,j)(i, j). In particular, ∫dx∫dy  ϕi,j(x,y)=h2;∫dx∫dy  ∇ϕi,j⋅∇ϕi,j=83;∫dx∫dy  ∇ϕi+1,j⋅∇ϕi,j=−13;∫dx∫dy  ∇ϕi,j+1⋅∇ϕi,j=−13;∫dx∫dy  ∇ϕi+1,j+1⋅∇ϕi,j=−13\begin{aligned} \int dx\int dy\;\phi_{i,j}(x, y) &= h^2; & \int dx\int dy\;\mathbf{\nabla}\phi_{i,j}\cdot\mathbf{\nabla}\phi_{i,j} &= \frac{8}{3};\\ \int dx\int dy\;\mathbf{\nabla}\phi_{i+1,j}\cdot\mathbf{\nabla}\phi_{i,j} &= -\frac{1}{3}; & \int dx\int dy\;\mathbf{\nabla}\phi_{i,j+1}\cdot\mathbf{\nabla}\phi_{i,j} &= -\frac{1}{3};\\ \int dx\int dy\;\mathbf{\nabla}\phi_{i+1,j+1}\cdot\mathbf{\nabla}\phi_{i,j} &= -\frac{1}{3} \end{aligned}

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