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This study introduces direct conservative domains (DCDs) to ensure local mass conservation in finite element methods. DCDs enable direct flux computation, improving conservation properties for numerical simulations.

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Area of Science:

  • Numerical Analysis
  • Computational Fluid Dynamics
  • Finite Element Methods

Background:

  • The continuous Galerkin finite element method (FEM) often exhibits local non-conservation issues.
  • Existing literature suggests postprocessing fluxes can achieve local conservation in FEM.
  • Direct flux computation from potential distribution typically leads to discontinuous fluxes and mass imbalance.

Purpose of the Study:

  • To propose the concept of a direct conservative domain (DCD) for achieving local mass conservation.
  • To develop a method for modifying advection fluxes to create various conservative domains.
  • To provide a theoretical basis for analyzing the local conservation of postprocessing algorithms using DCDs.

Main Methods:

  • Introduction of the direct conservative domain (DCD) concept.
  • Development of a flux modification technique for advection terms.
  • Application of DCDs to analyze and verify local conservation properties of FEM algorithms.

Main Results:

  • DCDs demonstrate the ability to conserve mass when fluxes are computed directly.
  • The proposed flux modification method generates different conservative domains.
  • DCDs provide a theoretical framework for evaluating postprocessing conservation algorithms.

Conclusions:

  • The direct conservative domain (DCD) concept offers a novel approach to local mass conservation in FEM.
  • The proposed method and DCDs are validated through a hypothetical 2-D model, confirming their effectiveness.
  • This work establishes a theoretical foundation for understanding and improving conservation in numerical methods.