The best method to solve Helmholtz equation for a irregular boundary

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SUMMARY

The best method to solve the Helmholtz equation for an irregular boundary is through the application of finite element methods, particularly when combined with Taylor series expansions to modify boundary conditions. This approach allows for the effective handling of curvy edges by approximating them with a square boundary. The reference to "Applied Numerical Methods" by Carnahan, Luther, and Wilkes provides foundational insights into this numerical technique.

PREREQUISITES
  • Understanding of the Helmholtz equation and its applications
  • Familiarity with Dirichlet boundary conditions
  • Basic knowledge of finite element methods
  • Proficiency in Taylor series expansions
NEXT STEPS
  • Research finite element method implementations in MATLAB or Python
  • Study Taylor series applications in numerical analysis
  • Explore boundary condition modifications in numerical simulations
  • Read "Applied Numerical Methods" by Carnahan, Luther, and Wilkes for deeper insights
USEFUL FOR

Mathematicians, engineers, and computational scientists involved in solving partial differential equations, particularly those working with irregular geometries and boundary conditions.

wdlang
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i have an almost square region.

By 'almost' i mean the edges are curvy, not completely straight.

i now need to solve the Helmholtz equation with Dirichlet boundary condition

what is the best numerical method?

how is Finite element, though i do not know what Finite element is
 
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By proper use of Taylor series expansions at the boundary, you can express a modified boundary condition on a square boundary (taking into account the difference between the actual boundary location and the square boundary location). This will allow you to use finite differences with the square boundary. See Carnahan, Luther, and Wilkes, Applied Numerical Methods.
 

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