On the finite element approximation for non-stationary saddle-point problems

TL;DR

This paper provides a rigorous error analysis for finite element approximations of non-stationary saddle-point problems, including hydrostatic Stokes equations, highlighting pressure error estimates with singularity considerations.

Contribution

It establishes new error estimates for finite element solutions of non-stationary saddle-point problems, incorporating the effects of semigroup analyticity on pressure errors.

Findings

Derived error estimates for finite element solutions in various norms.

Identified a natural $t^{-1}$ singularity in pressure error due to semigroup analyticity.

Applied error estimates to hydrostatic Stokes equations relevant for geophysical flows.

Abstract

In this paper, we present a numerical analysis of the hydrostatic Stokes equations, which are linearization of the primitive equations describing the geophysical flows of the ocean and the atmosphere. The hydrostatic Stokes equations can be formulated as an abstract non-stationary saddle-point problem, which also includes the non-stationary Stokes equations. We first consider the finite element approximation for the abstract equations with a pair of spaces under the discrete inf-sup condition. The aim of this paper is to establish error estimates for the approximated solutions in various norms, in the framework of analytic semigroup theory. Our main contribution is an error estimate for the pressure with a natural singularity term $t^{-1}$, which is induced by the analyticity of the semigroup. We also present applications of the error estimates for the finite element approximations of the non-stationary Stokes and the hydrostatic Stokes equations.