<p>In this paper, we present a novel, efficient, and uniformly stable Crouzeix–Raviart finite element method for the Brinkman problem. The new formulation is based on decomposition of the bilinear form associated with the Darcy term into a sum of two bilinear forms: one corresponding to a projected velocity and the other to its fluctuation. The resulting method, robust across all material parameters, is symmetric and stable without requiring any stability constant. Optimal error estimates are derived under minimal regularity assumptions on the solution, and the velocity error shown to be independent of the pressure. Additionally, we introduce a simple postprocessing method for the velocity to ensure normal flux continuity of the velocity across interelement boundaries. Several numerical experiments, including a practical example of the Brinkman flow with actual material parameters from the SPE10 dataset, demonstrate the effectiveness of the proposed method.</p>

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A parameter-robust Crouzeix–Raviart finite element method for the Brinkman model

  • Eun-Jae Park,
  • Hyeokjoo Park

摘要

In this paper, we present a novel, efficient, and uniformly stable Crouzeix–Raviart finite element method for the Brinkman problem. The new formulation is based on decomposition of the bilinear form associated with the Darcy term into a sum of two bilinear forms: one corresponding to a projected velocity and the other to its fluctuation. The resulting method, robust across all material parameters, is symmetric and stable without requiring any stability constant. Optimal error estimates are derived under minimal regularity assumptions on the solution, and the velocity error shown to be independent of the pressure. Additionally, we introduce a simple postprocessing method for the velocity to ensure normal flux continuity of the velocity across interelement boundaries. Several numerical experiments, including a practical example of the Brinkman flow with actual material parameters from the SPE10 dataset, demonstrate the effectiveness of the proposed method.