TY - JOUR
T1 - On the initial higher-order pressure convergence in equal-order finite element discretizations of the Stokes system
AU - Pacheco, Douglas R.Q.
AU - Steinbach, Olaf
N1 - Publisher Copyright:
© 2022 The Authors
PY - 2022/3/1
Y1 - 2022/3/1
N2 - In incompressible flow problems, the finite element discretization of pressure and velocity can be done through either stable spaces or stabilized pairs. For equal-order stabilized methods with piecewise linear discretization, the classical theory guarantees only linear convergence for the pressure approximation. However, a higher order is often observed, yet seldom discussed, in numerical practice. Such experimental observations may, in the absence of a sound a priori error analysis, mislead the selection of finite element spaces in practical applications. Therefore, we present here a numerical analysis demonstrating that an initial higher-order pressure convergence may in fact occur under certain conditions, for equal-order elements of any degree. Moreover, our numerical experiments clearly indicate that whether and for how long this behavior holds is a problem-dependent matter. These findings confirm that an optimal pressure convergence can in general not be expected when using unbalanced velocity-pressure pairs.
AB - In incompressible flow problems, the finite element discretization of pressure and velocity can be done through either stable spaces or stabilized pairs. For equal-order stabilized methods with piecewise linear discretization, the classical theory guarantees only linear convergence for the pressure approximation. However, a higher order is often observed, yet seldom discussed, in numerical practice. Such experimental observations may, in the absence of a sound a priori error analysis, mislead the selection of finite element spaces in practical applications. Therefore, we present here a numerical analysis demonstrating that an initial higher-order pressure convergence may in fact occur under certain conditions, for equal-order elements of any degree. Moreover, our numerical experiments clearly indicate that whether and for how long this behavior holds is a problem-dependent matter. These findings confirm that an optimal pressure convergence can in general not be expected when using unbalanced velocity-pressure pairs.
KW - Finite element methods
KW - Incompressible flow
KW - Numerical analysis
KW - Stabilized finite element methods
KW - Stokes system
KW - Superconvergence
UR - http://www.scopus.com/inward/record.url?scp=85123698049&partnerID=8YFLogxK
U2 - 10.1016/j.camwa.2022.01.022
DO - 10.1016/j.camwa.2022.01.022
M3 - Article
AN - SCOPUS:85123698049
SN - 0898-1221
VL - 109
SP - 140
EP - 145
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
ER -