Direct Conservative Domain in the Continuous Galerkin Method for Groundwater Models

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.