Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Differential Geometry

Background:

• Solving partial differential equations (PDEs) on curved surfaces presents significant computational challenges.
• Existing numerical methods often require explicit surface meshing or approximations, which can introduce errors.
• The Virtual Element Method (VEM) offers a flexible framework for discretizing complex geometries.

Purpose of the Study:

• To develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) for elliptic surface PDEs.
• To enable accurate numerical solutions without explicit surface geometry approximation.
• To extend the theoretical properties of VEM to discretizations on surfaces.

Main Methods:

• Formulation of elliptic surface PDEs in covariant form using local reference systems.
• Application of a two-dimensional VEM scheme leveraging local parametrization.
• Extension of classical VEM theoretical properties to handle anisotropic discretizations.

Main Results:

• Demonstrated the effectiveness of the geometrically intrinsic VEM formulation on polygonal cells.
• Validated theoretical properties through extensive testing on triangular and polygonal meshes with manufactured solutions.
• Identified limitations related to surface regularity and approximation accuracy.

Conclusions:

• The proposed VEM formulation provides a robust and accurate method for solving PDEs on surfaces.
• The intrinsic approach avoids explicit surface approximation, simplifying the numerical solution process.
• Further research can explore the method's performance on more complex surfaces and PDE types.