Research

In this thesis, we have utilized the Galerkin finite element method to solve the Helmholtz equation. We have applied various combinations of domain decompositions with different types of two-dimensional elements. We have emphasised on quadrilateral elements, especially on eight-noded and nine-noded elements. The results obtained by quadrilateral elements are better than those of the triangular elements. Again, eight-noded and nine-noded elements have provided improved results than the four-noded elements. It is also highlighted that the method involving nine-noded elements gave better results than the eight-noded elements in almost every case. It is very amazing that the finite element technique performs well in the case of a curved side domain. It takes some information at some points on the boundary of the domain and produces the approximations at each and every point in the whole domain, like an analytical solution. It proves that it is really an important tool for numerical analysis. Here we have not worked more with the triangular elements. We have just added a single example with three-node triangular elements. We hope that the six-noded triangular element will provide good results. We have an intention to work with this type of quadratic element. It is observed that if we have increased the number of elements successively, then the result has improved accordingly. But after a certain increment, the results did not converge. So, there arises an issue to analyse the stability criteria for every element. Besides these, there are also some related topics to our work for further study.