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<p>Incompressible flows are modeled by a coupled system of partial differential equations for velocity and pressure. Starting from a divergence-free mixed method proposed in [John, Li, Merdon and Rui, Math. Models Methods Appl. Sci. 34(05):919–949, 2024], this paper proposes <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{H}(\textrm{div})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">H</mi> </mrow> <mo stretchy="false">(</mo> <mtext>div</mtext> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-conforming finite element methods which decouple the velocity and pressure by constructing divergence-free basis functions. The methods are derived and analyzed for the Stokes problem. Algorithmic issues like the computation of this basis and the imposition of non-homogeneous Dirichlet boundary conditions are discussed. Numerical studies on two- and three-dimensional Stokes problems compare the efficiency of the proposed methods with methods from the above mentioned paper.</p>

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Divergence-Free Decoupled Finite Element Methods for Incompressible Flow Problems

  • 期刊论文

  • 发表日期：

    2026年6月16日

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  • Volker John,
  • Xu Li,
  • Christian Merdon

摘要

Incompressible flows are modeled by a coupled system of partial differential equations for velocity and pressure. Starting from a divergence-free mixed method proposed in [John, Li, Merdon and Rui, Math. Models Methods Appl. Sci. 34(05):919–949, 2024], this paper proposes \(\varvec{H}(\textrm{div})\) H ( div ) -conforming finite element methods which decouple the velocity and pressure by constructing divergence-free basis functions. The methods are derived and analyzed for the Stokes problem. Algorithmic issues like the computation of this basis and the imposition of non-homogeneous Dirichlet boundary conditions are discussed. Numerical studies on two- and three-dimensional Stokes problems compare the efficiency of the proposed methods with methods from the above mentioned paper.