Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions

Abstract We present in this paper a rigorous error analysis of the vector penalty-projection method for solving the time-dependent incompressible Stokes equations with open boundary conditions on part of the boundary. First, we prove the stability of the scheme. Then we provide an error analysis for the second-order vector penalty-projection method which shows that the convergence rate of the error on the velocity and the pressure is of order 2 in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>l</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi mathvariant="bold">L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> l^{\infty}(\mathbf{L}^{2}(\Omega)) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>l</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> l^{2}(L^{2}(\Omega)) respectively. In addition, it is shown that the splitting errors of the method varies as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">O</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ε</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{O}(\varepsilon) , where 𝜀 is a penalty parameter chosen as small as desired. Several numerical tests in agreement with the theoretical results are presented. To the best of our knowledge, this paper provides the first rigorous proof of optimal error estimates for second-order splitting schemes with open boundary conditions.