Conservation of Forces and Total Work at the Interface Using the Internodes Method.

Area of Science:

• Numerical analysis
• Computational science
• Scientific computing

Background:

• Non-conforming discretizations of partial differential equations (PDEs) pose challenges in maintaining solution conservation across subdomain interfaces.
• The Internodes method offers a general approach for such problems in 2D and 3D regions.

Purpose of the Study:

• To quantify the conservation properties of the Internodes method across interfaces for non-conforming discretizations.
• To analyze the behavior of conservation under hp-fem discretizations as mesh size tends to zero.

Main Methods:

• Theoretical analysis of conservation properties for hp-fem discretizations.
• Focus on second-order elliptic PDEs, using terminology from linear elasticity (forces and works).
• Numerical experiments in 2D and 3D to validate theoretical findings.

Main Results:

• Proved that total force and work at the interface vanish optimally, decaying as O(h^{p+1}) for hp-fem.
• Demonstrated that conservation decay matches the error decay in the H^1-broken norm.
• Showcased that conservation is intrinsic to the method's interface condition enforcement.

Conclusions:

• The Internodes method exhibits optimal conservation properties across subdomain interfaces.
• Conservation is independent of the specific PDE problem, depending solely on the discretization method.
• Numerical results confirm theoretical findings and compare favorably with Mortar and WACA methods.