A discontinuous Galerkin pressure correction scheme for the Oldroyd model of order one

TL;DR

This paper introduces a novel discontinuous Galerkin pressure correction scheme for the Oldroyd model of order one, providing theoretical analysis and numerical validation of stability, existence, uniqueness, and optimal error bounds.

Contribution

The paper develops a new discontinuous Galerkin scheme for the Oldroyd model, with rigorous proofs of stability, convergence, and optimal error estimates, including improved velocity error bounds.

Findings

Proved existence and uniqueness of the discrete solution.

Established stability of velocity and pressure.

Confirmed optimal convergence rates through numerical experiments.

Abstract

We develop and analyze a discontinuous Galerkin pressure correction scheme for the Oldroyd model of order one. The existence and uniqueness of the discrete solution as well as the consistency of the scheme are proved. The stability of the discrete velocity and pressure are established. We derive optimal $\textit{a priori}$ error bounds for the fully discrete velocity in the discontinuous discrete space. In addition, an improved error estimate for the velocity is derived in the $L^2$ norm which is optimal with respect to space and time. Furthermore, the error bound for the pressure is obtained via the estimates of discrete time derivative of the velocity. Finally, numerical experiments confirm the optimal convergence rates.