An Optimal Finite Element Method with Uzawa Iteration for Stokes Equations including Corner Singularities

The Uzawa method is an iterative approach to find approximated solutions to the Stokes equations. This method solves velocity variables involving augmented Lagrangian operator and then updates pressure variable by Richardson update. In this paper, we construct a new version of the Uzawa method to find optimal numerical solutions of the Stokes equations including corner singularities. The proposed method is based on the dual singular function method which was developed for elliptic boundary value problems. We estimate the solvability of the proposed formulation and special orthogonality form for two singular functions. Numerical convergence tests are presented to verify our assertion.

Water quality model with axial dispersion solved by Eulerian-Lagrangian operator-splitting method in water distribution system

This study explores the effects of water quality simulation results by embedding axial dispersion into the classical advective-reactive model in a water distribution system. The Eulerian-Lagrangian operator-splitting method is employed to solve the model with axial dispersion. Satisfactory results were obtained after the numerical solution was tested against the analytical and other numerical solutions. The water quality simulation results without the reaction item show that when water velocity is low (or Peclet numbers are small), dispersion is dominant and significantly affects the numerical simulation results. The contaminant concentration at downstream node gradually increased with time along the upstream pipelines from the source, which is particularly reflected in the terminal. The simulation results show that the biomass concentration may demonstrate synthetic effects of axial dispersion and reaction, i.e. mutual promotion, given the multicomponent (substrate, residual chlorine, and microbial biomass) reaction-transport processes. It is particularly reflected in the low flow velocity.