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leon-vv / Traceon

Electromagnetic solver and electron tracer
https://traceon.org/
GNU Affero General Public License v3.0
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Research using higher order approximations at the nodes/line elements #8

Closed leon-vv closed 1 year ago

leon-vv commented 1 year ago

Quoting from Peter W. Hawkes. Pinciples of Electron Optics, Basic Geometrical Optics. 2017.

Section 10.2.1:

Another advantage is the possibility of using more than the minimum number N of trial functions and variables Q k . A favourable choice is, for instance, the use of function values Y(s n ) and derivatives Y’(s n ) at the nodes, hence M = 2N. In each boundary element the function Y(s) can then be approximated by a cubic Hermite polynomial, which is fairly accurate. Each node has two degrees of freedom, which gives a better result than the simple one-degree approximation.

leon-vv commented 1 year ago

I think it would be optimal (can we prove this?) to choose the points at which the charge is solved as the nodes of a Gaussian quadrature

leon-vv commented 1 year ago

Implemented for radial symmetry in v0.2.0. Seems to work well. Implementation started on 3D but it seems to perform poorly if not combined with higher order line elements as well.

Made separate issues: #54 #55