## Abstract

Based on a fully overlapping domain decomposition technique, a parallel stabilized equal-order finite element method for the steady Stokes equations is presented and studied. In this method, each processor computes a local stabilized finite element solution in its own subdomain by solving a global problem on a global mesh that is locally refined around its subdomain, where the lowest equal-order finite element pairs (continuous piecewise linear, bilinear or trilinear velocity and pressure) are used for the finite element discretization and a pressure-projection-based stabilization method is employed to circumvent the discrete inf–sup condition that is invalid for the used finite element pairs. The parallel stabilized method is unconditionally stable, free of parameter and calculation of derivatives, and is easy to implement based on an existing sequential solver. Optimal error estimates are obtained by the theoretical tool of local a priori error estimates for finite element solutions. Numerical results are also given to verify the theoretical predictions and illustrate the effectiveness of the method.

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This work was supported by the Natural Science Foundation of China (No. 11361016), the Basic and Frontier Explore Program of Chongqing Municipality, China (Nos. cstc2016jcyjA0348, cstc2018jcyjAX0305), and Fundamental Research Funds for the Central Universities (No. XDJK2018B032).

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Shang, Y. A parallel stabilized finite element method based on the lowest equal-order elements for incompressible flows.
*Computing* **102, **65–81 (2020). https://doi.org/10.1007/s00607-019-00729-0

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### Keywords

- Incompressible flow
- Stokes equations
- Finite element
- Stabilized method
- Parallel algorithm
- Domain decomposition

### Mathematics Subject Classification

- 35Q30
- 65N30
- 65N55
- 68W10
- 76D07