Journal
ADVANCES IN COMPUTATIONAL MATHEMATICS
Volume 44, Issue 1, Pages 195-225Publisher
SPRINGER
DOI: 10.1007/s10444-017-9540-1
Keywords
Incompressible Navier-Stokes equations; Inf-sup stable finite element methods; Grad-div stabilization; Error bounds independent of the viscosity; Nonlocal compatibility condition; Backward Euler method
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Funding
- Spanish MINECO [MTM2013-42538-P, MTM2015-65608-P, MTM2016-78995-P]
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This paper studies inf-sup stable finite element discretizations of the evolutionary Navier-Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier-Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Both the continuous-in-time case and the fully discrete scheme with the backward Euler method as time integrator are analyzed.
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