<p>We analyze a second-order projection method for the incompressible Navier-Stokes equations on bounded Lipschitz domains. The scheme employs a Backward Differentiation Formula of order two (BDF2) for the time discretization, combined with conforming finite elements in space. Projection methods are widely used to enforce incompressibility, yet rigorous convergence results for possibly non-smooth solutions have so far been restricted to first-order schemes. We establish, for the first time, convergence (up to subsequence) of a second-order projection method to Leray–Hopf weak solutions under minimal assumptions on the data, namely <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u_0 \in L^2_{\textrm{div}}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>∈</mo> <msubsup> <mi>L</mi> <mtext>div</mtext> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f \in L^2(0,T;L^2_{\textrm{div}}(\Omega ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msubsup> <mi>L</mi> <mtext>div</mtext> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our analysis relies on two ingredients: A discrete energy inequality providing uniform <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^\infty (0,T;L^2(\Omega ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^2(0,T;H^1_0(\Omega ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> bounds for suitable interpolants of the discrete velocities, and a compactness argument combining Simon’s theorem with refined time-continuity estimates. These tools overcome the difficulty that only the projected velocity satisfies an approximate divergence-free condition, while the intermediate velocity is controlled in space. We conclude that a subsequence of the approximations converges to a Leray–Hopf weak solution. This result provides the first rigorous convergence proof for a higher-order projection method under no additional assumptions on the solution beyond those following from the standard a priori energy estimate.</p>

Convergence of a Second-Order Projection Method to Leray-Hopf Solutions of the Incompressible Navier-Stokes Equations

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2026年3月19日

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Franziska Weber

摘要

We analyze a second-order projection method for the incompressible Navier-Stokes equations on bounded Lipschitz domains. The scheme employs a Backward Differentiation Formula of order two (BDF2) for the time discretization, combined with conforming finite elements in space. Projection methods are widely used to enforce incompressibility, yet rigorous convergence results for possibly non-smooth solutions have so far been restricted to first-order schemes. We establish, for the first time, convergence (up to subsequence) of a second-order projection method to Leray–Hopf weak solutions under minimal assumptions on the data, namely $u_0 \in L^2_{\textrm{div}}(\Omega )$ $u_{0} \in L_{div}^{2} (Ω)$ and $f \in L^2(0,T;L^2_{\textrm{div}}(\Omega ))$ $f \in L^{2} (0, T; L_{div}^{2} (Ω))$ . Our analysis relies on two ingredients: A discrete energy inequality providing uniform $L^\infty (0,T;L^2(\Omega ))$ $L^{\infty} (0, T; L^{2} (Ω))$ and $L^2(0,T;H^1_0(\Omega ))$ $L^{2} (0, T; H_{0}^{1} (Ω))$ bounds for suitable interpolants of the discrete velocities, and a compactness argument combining Simon’s theorem with refined time-continuity estimates. These tools overcome the difficulty that only the projected velocity satisfies an approximate divergence-free condition, while the intermediate velocity is controlled in space. We conclude that a subsequence of the approximations converges to a Leray–Hopf weak solution. This result provides the first rigorous convergence proof for a higher-order projection method under no additional assumptions on the solution beyond those following from the standard a priori energy estimate.