Enriched finite elements for time-harmonic Webster's equation

This work presents an enriched finite element method (FEM) dedicated to the numerical resolution of Webster's equation in the time-harmonic regime, which models many physical configurations, e.g. wave propagation in acoustic waveguides or vibration of bars with varying cross-section. Building on the wave-based methods existing in the literature, we present new enriched finite element bases that account for both the frequency of the problem and the heterogeneity of the coefficients of the equation. The enriched method is compared to the classical fifth-order polynomial FEM, and we show they share the same asymptotic convergence order, present the same easiness of implementation and have similar computational costs. The main improvement brought by these enriched bases relies on the convergence threshold (the mesh size at which the convergence regime begins) and the convergence multiplicative constant, which are observed to be (i) better than the ones associated with polynomial bases and (ii) not only dependent on the resolution (the number of elements per wavelength), but also on the frequency for a fixed resolution, making the method we propose well adapted to high-frequency regimes. Moreover, taking into account the heterogeneity of the coefficient in Webster's equation by using an element-dependent enrichment leads to a significant decrease of the approximation error on the considered examples compared to a uniform enrichment, again with almost no additional cost. Several possible extensions of this work are finally discussed.