A non‐conforming least‐squares finite element method for incompressible fluid flow problems

Abstract

SUMMARYIn this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C0 elements for the vorticity and the pressure. The new method, which we term dV‐VP improves upon our previous discontinuous stream‐function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second‐order terms from the least‐squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one‐half of the dimension of a stream‐function element of equal accuracy. In two dimensions, the discontinuous stream‐function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV‐VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA.

A Sequential Least Squares Method for Poisson Equation Using a Patch Reconstructed Space

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A conservative stable finite element method for Stokes flow and nearly incompressible linear elasticity on rectangular grid

“…We remark that, there have been various approaches utilised in regard to the conditions, such as the discontinuous Galerkin methods [24,[28][29][30], the isogeometric methods [33][34][35], the least square finite element methods [11,14,26,43], finite element methods with enhanced stabilisation [9,13,23,25,62], the element free Galerkin method [10,67], stabilised finite element with postprocessing [8], and the NURBS method [6,50]. They provide various approaches to construct a divergence-free velocity field for Stokes problems.…”

A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition

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