Behavior of the conditioning number in the Ultra-Weak Variational Formulation, with Bessel functions as a basis for the non-homogeneous Helmholtz equation
DOI:
https://doi.org/10.35819/remat2024v10iespecialid7054Keywords:
ultra weak variational formulation, Helmholtz equation, ill-conditioned systems, cylindrical waves, Bessel functionsAbstract
The Ultra Weak Variational Formulation (UWVF) emerges as a promising methodology for simulating various wave phenomena. However, the linear system resulting from the discretization of this formulation can be highly ill-conditioned, compromising strategies for estimating the error of the approximate solution. In this research, we analyze the conditioning of the underlying linear system concerning the choice of certain families of basis functions, including classical plane waves, in the UWVF applied to a boundary value problem (BVP) for the Helmholtz equation. Among the implemented families, the one formed by cylindrical waves based on Bessel functions, scaled by a global factor also based on Bessel functions, stood out for implying a significantly lower conditioning number than those produced by other families. This prominence was observed in all numerical experiments conducted, both for the case of the homogeneous Helmholtz equation and for the non-homogeneous case, while varying both the number of functions in the family in question and the refinement of the computational meshes used.
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