Application of the residual-free bubbles to the Helmhotz equation with large wave numbers and a fourth order method with seven-point stencil

A new two-level finite element method is introduced for the approximations of the residual-free bubble (RFB) functions and its application to the Helmholtz equation with large wave numbers is considered. Altough this approach was considered for the Helmholtz equation before, our new insights show that some of its important properties have remained hidden. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Furthermore, it is possible to further increase the accuracy of the solutions in 2D with a simple modification of the RFB method. We provide analysis to show how the modified-RFB method mitigates the pollution error. The analysis gave rise to a fourth order accurate finite difference scheme that uses seven-point stencil. The modified-RFB is able to solve the Helmholtz equation efficiently in 2D up to ch = 3.5 where c is the wave number and h is the mesh size.