# Advanced Numerical Methods in the Mathematical Sciences

## Subhashree Mohapatra, Indian Institute of Science at Bangalore (India) Least Squares Spectral Element Method for Oseen Equations with Applications to Navier-Stokes Equations

### Authors

• Subhashree Mohapatra, Indian Institute of Science at Bangalore (India)
• Sashikumaar Ganesan, Indian Institute of Science at Bangalore (India)

### Abstract

We propose a non-conforming least-squares spectral element method for Oseen equations in primitive variable formulation for both two- and three-dimensional domains. In the proposed numerical scheme, the first order reformulation is avoided. Further, the condition number of algebraic system is controlled by a suitable choice of preconditioner for the velocity components and the pressure variable in such a way that the condition number of the preconditioned system is $$(\ln{W})^{2}$$, $$W$$ being the degree of the polynomial. Moreover, the preconditioned conjugate gradient method is used to solve the system of algebraic equations. The proposed scheme is proved to be exponentially accurate. Also, the Navier-Stokes equations in primitive variable formulation is solved by solving a sequence of Oseen type equations. Numerical results of the Oseen equations and the Navier-Stokes equations in two-dimensional domains with different Reynolds number are presented. Similar order of convergence is achieved for all Reynolds number cases at the cost of more number of iteration for high Reynolds numbers.