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An adapted bubble approach which is a modifiation of the residual-freebubbles (RFB) method, is proposed for the Helmhotz problem in 2D. A newtwo-level finite element method is introduced for the approximations of thebubble functions. Unlike the other equations such as the advection-diffusionequation, RFB method when applied to the Helmholtz equation, does not depend onanother stabilized method to obtain approximations to the solutions of thesub-problems. Adapted bubbles (AB) are obtained by a simple modification of thesub-problems. This modification increases the accuracy of the numericalsolution impressively. The AB method is able to solve the Helmholtz equationefficiently in 2D up to ch = 3.5 where c is the wave number and h is the meshsize. We provide analysis to show how the AB method mitigates the pollutionerror.

Application of adapted-bubbles to the Helmholtz equation with large wave numbers in 2d