TY - JOUR

T1 - Analysis of the L2 least-squares finite element method for the velocity-vorticity-pressure Stokes equations with velocity boundary conditions

AU - Chang, Ching L.

AU - Yang, Suh Yuh

PY - 2002/7/25

Y1 - 2002/7/25

N2 - A theoretical analysis of the L2 least-squares finite element method (LSFEM) for solving the Stokes equations in the velocity-vorticity-pressure (VVP) first-order system formulation with the Dirichlet velocity boundary condition is given. The least-squares energy functional is defined to be the sum of the squared L2-norms of the residuals in the partial differential equations, weighted appropriately by the viscosity constant ν. It is shown that, with many advantages, the method is stable and convergent without requiring extra smoothness of the exact solution, and the piecewise linear finite elements can be used to approximate all the unknowns. Furthermore, with respect to the order of approximation for smooth exact solutions, the method exhibits an optimal rate of convergence in the H1-norm for velocity and a suboptimal rate of convergence in the L2-norm for vorticity and pressure. Some numerical experiments in two and three dimensions are given, which confirm the a priori error estimates. Since the boundary of the bounded domain under consideration is polygonal in ℛ2 or polyhedral in ℛ3 instead of C1-smooth, the authors adopt the more direct technique of Bramble-Pasciak and Cai-Manteuffel-McCormick, which departs from the Agmon-Douglis-Nirenberg theory, in showing the coercivity bound of the least-squares functional.

AB - A theoretical analysis of the L2 least-squares finite element method (LSFEM) for solving the Stokes equations in the velocity-vorticity-pressure (VVP) first-order system formulation with the Dirichlet velocity boundary condition is given. The least-squares energy functional is defined to be the sum of the squared L2-norms of the residuals in the partial differential equations, weighted appropriately by the viscosity constant ν. It is shown that, with many advantages, the method is stable and convergent without requiring extra smoothness of the exact solution, and the piecewise linear finite elements can be used to approximate all the unknowns. Furthermore, with respect to the order of approximation for smooth exact solutions, the method exhibits an optimal rate of convergence in the H1-norm for velocity and a suboptimal rate of convergence in the L2-norm for vorticity and pressure. Some numerical experiments in two and three dimensions are given, which confirm the a priori error estimates. Since the boundary of the bounded domain under consideration is polygonal in ℛ2 or polyhedral in ℛ3 instead of C1-smooth, the authors adopt the more direct technique of Bramble-Pasciak and Cai-Manteuffel-McCormick, which departs from the Agmon-Douglis-Nirenberg theory, in showing the coercivity bound of the least-squares functional.

KW - A priori error estimates

KW - Condition numbers

KW - Convergence

KW - Least-squares finite element methods

KW - Stability

KW - Velocity-vorticity-pressure Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=0037173504&partnerID=8YFLogxK

U2 - 10.1016/S0096-3003(01)00086-8

DO - 10.1016/S0096-3003(01)00086-8

M3 - 期刊論文

AN - SCOPUS:0037173504

SN - 0096-3003

VL - 130

SP - 121

EP - 144

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

IS - 1

ER -