A posteriori error estimates of stabilized low-order mixed finite elements for the Stokes eigenvalue problem

A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

“…It is well known that the inf-sup condition holds for the continuous problem (6), and the corresponding solution operator is compact. From the spectral theory ( [8]) it follows that (6) has a sequence of real eigenvalues (see also [2,4,15]) which are assumed to satisfy…”

“…A sample discretization of the problem domain using N = 5 is illustrated in Figure 1. As we have already mentioned, the exact solution is unknown, and we take λ 1 = 52.3447 as a reference to the minimum eigenvalue (see [2,4,20]). Figures 2 and 3 present the convergence of the minimum eigenvalue approximations to the reference value λ 1 for the two-field and three-field problems, re- spectively.…”

“…where L * is the formal adjoint operator of L , and α K is a stabilization matrix (if u h is vector valued) of numerical parameters defined within each element domain K. Here and in the following, K stands for the summation over all elements of the finite element partition, and (·, ·) K for the L 2 (K)-inner product. It is clear from (4) that in general the resulting system leads to a quadratic eigenvalue problem, which, apart from being much more demanding than a linear one, could introduce eigenpairs that converge to solutions which are not solutions of the original problem (2).…”

Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

“…Some superconvergence results and the related recovery type a posteriori error estimators for the Stokes eigenvalue problem is presented by Liu et al [25] based on a projection method. In [2], Armentano et al introduced a posteriori error estimators for stabilized low-order mixed finite elements and in [16], Han et al presented a residual type a posterior error estimator for a new adaptive mixed finite element method for the Stokes eigenvalue problem. In [18], Huang presents a posteriori lower and upper eigenvalue bounds for the Stokes eigenvalue problem for two stabilized finite element methods based on the lowest equal-order finite element pair.…”

Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem based on penalty techniques

“…There are numerous works devoted to approximating the Stokes eigenproblem based on different methodologies. Among them are finite element methods [1,8,12,18], mesh free methods based on radial basis functions [6], spectral Chebyshev methods based on decoupling the velocity and pressure operators [10,11], and spectral Lagrange method using a staggered grid system [4].…”