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01-09-2023

A macro-bubble enriched P1–P0 finite element for the Stokes equations on triangular and tetrahedral meshes

Author: Shangyou Zhang

Published in: Calcolo | Issue 3/2023

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Abstract

The Bernardi–Raugel finite element is a bubble enriched \(P_1\)-\(P_0\) finite element for the Stokes equations, where three \(P_2\) edge-bubbles and four \(P_3\) face-bubbles are added to the \(P_1\) velocity space on each triangle and each tetrahedron respectively. In the new macro-bubble enriched finite element method, the velocity space is also enriched by three \(P_2\) but multi-piece bubbles on each triangle. The divergence of such a \(P_2\) bubble function is not a piecewise \(P_1\) function but a one-piece \(P_0\) function on the triangle. In 3D, four \(P_1\) macro-bubbles are added to the \(P_1\) velocity space where the divergence of such a macro-bubble function is also a one-piece \(P_0\) function. The macro-bubble enriched P1–P0 finite element is shown stable and quasi-optimal in solving the Stokes equations. Additionally the method is shown viscosity robust that the accuracy of the discrete solutions is independent of viscosity, neither the smoothness of exact pressure solution. Numerical tests show the new method is equally good for small Reynolds number flows, but superior to the existing method for flows with large Reynolds numbers.

previous article Conditioning of Fourier sums on a quadratic curve

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Title: A macro-bubble enriched P1–P0 finite element for the Stokes equations on triangular and tetrahedral meshes
Author: Shangyou Zhang
Publication date: 01-09-2023
Publisher: Springer International Publishing
Published in: Calcolo / Issue 3/2023
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-023-00538-0

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A macro-bubble enriched P1–P0 finite element for the Stokes equations on triangular and tetrahedral meshes

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