Wind turbine wake aerodynamics

We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier--Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.

/pdf/finite-element-methods-of-least-squares-type-1dl5v7wlyu.pdf

Finite Element Methods of Least-Squares Type

Unsteady solution of incompressible Navier-Stokes equations

An overview is given of new developments of the least squares finite element method (LSFEM) in fluid dynamics. Special emphasis is placed on the universality of LSFEM; the symmetry and positiveness of the algebraic systems obtained from LSFEM; the accommodation of LSFEM to equal order interpolations for incompressible viscous flows; and the natural numerical dissipation of LSFEM for convective transport problems and high speed compressible flows. The performance of LSFEM is illustrated by numerical examples.

/pdf/least-squares-finite-element-method-for-fluid-dynamics-26utf9nl6m.pdf

Least-squares finite element method for fluid dynamics

A numerical study of the two-dimensional Navier-Stokes equations in vorticity-velocity variables

The numerical solution of the Navier-Stokes equations for 3-dimensional, unsteady, incompressible flows by compact schemes

A least squares formulation of the system divu = rho, curlu = zeta is surveyed from the viewpoint of both finite element and finite difference methods. Closely related arguments are shown to establish convergence estimates.

/pdf/a-comparative-study-of-finite-element-and-finite-difference-vsdmrejtl3.pdf

A comparative study of finite element and finite difference methods for cauchy-riemann type equations*

This paper discusses a class of compact second order accurate finite difference equations for mixed initial-boundary value problems for hyperbolic and convective-diffusion equations Convergence is proved by means of energy arguments and both types of equations are solved by similar algorithms For hyperbolic equations an extension of the Lax–Wendroff method is described which incorporates dissipative boundary conditions Upwind-downwind differencing techniques arise as the formal hyperbolic limit of the convective-diffusion equation Finally, a finite difference “chain-rule” transforms the schemes from rectangular to quadrilateral subdomains

Compact Finite Difference Schemes for Mixed Initial-Boundary Value Problems

A ‘unified’ second-order accurate finite difference approach to treating the wave equation and the Cauchy’Riemann equations is developed. The procedure is motivated by an analysis of properties of weak solutions.

A `Unified' Numerical Treatment of the Wave Equation and the Cauchy--Riemann Equations

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