An H(div)-conforming Finite Element Method for Biot's Consolidation Model.

Area of Science:

• Computational mechanics
• Geotechnical engineering
• Numerical analysis

Background:

• Biot's consolidation model is crucial for understanding fluid flow and deformation in porous media.
• Existing numerical methods may face challenges with H(div)-conformity for displacement approximation.
• Poroelasticity simulations require robust and accurate discretization techniques.

Purpose of the Study:

• To develop a novel H(div)-conforming finite element method for Biot's consolidation model.
• To relax the H(div)-conformity requirement for displacement approximation.
• To analyze the theoretical properties and convergence of the proposed numerical scheme.

Main Methods:

• Discretization of flow variables using H(div)-conforming mixed finite elements.
• Approximation of displacement using H(div)-conforming finite elements with interior penalty discontinuous Galerkin for tangential components.
• Mathematical analysis to prove existence and uniqueness theorems for semi-discrete and fully discrete schemes.

Main Results:

• Successful development of an H(div)-conforming finite element method for Biot's model.
• Demonstration of relaxed H(div)-conformity for displacement approximation.
• Derivation of optimal convergence rates for both semi-discrete and fully discrete solutions.

Conclusions:

• The proposed H(div)-conforming finite element method offers a robust approach for Biot's consolidation model.
• The method provides theoretical guarantees for the existence, uniqueness, and accuracy of approximate solutions.
• This work advances numerical techniques in poroelasticity and computational geomechanics.