The aim of this thesis is to develop a flow solver that has the ability to obtain an accurate numerical solution fast and efficiently with minimum user intervention. In this study, a two-dimensional viscous, laminar, incompressible flow solver based on Least-Squares Spectral Element Method (LSSEM) is developed. The LSSEM flow solver can work on hp-type nonconforming grids and can perform p-type adaptive refinement. Several benchmark problems are solved in order to validate the solver and successful results are obtained. In particular, it is demonstrated that p-type adaptive refinement on hp-type non-conforming grids can be used to improve the quality of the solution. Moreover, it is found that mass conservation performance of LSSEM can be enhanced by using p-type adaptive refinement strategies while keeping computational costs reasonable.

The main purpose of this study is to develop a flow solver that employs triangular grids to solve two-dimensional, viscous, laminar, steady, incompressible flows. The flow solver is based on Least Squares Spectral Element Method (LSSEM). It has p-type adaptive mesh refinement/coarsening capability and supports p-type nonconforming element interfaces. To validate the developed flow solver several benchmark problems are studied and successful results are obtained. The performances of two different triangular nodal distributions, namely Lobatto distribution and Fekete distribution, are compared in terms of accuracy and implementation complexity. Accuracies provided by triangular and quadrilateral grids of equal computational size are compared. Adaptive mesh refinement studies are conducted using three different error indicators, including a novel one based on elemental mass loss. Effect of modifying the least-squares functional by multiplying the continuity equation by a weight factor is investigated in regards to mass conservation.