Weak Galerkin finite element method for second order problems on curvilinear polytopal meshes with Lipschitz...

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Finite Element Methods

Background:

• Traditional finite element methods often struggle with complex geometries and less smooth boundaries.
• Curvilinear elements and polytopal meshes present challenges for standard numerical techniques.
• The weak Galerkin (WG) finite element method offers a flexible framework but requires robust basis functions for complex domains.

Purpose of the Study:

• To develop and analyze new basis functions for the weak Galerkin finite element method tailored for curvilinear elements.
• To enable the accurate numerical solution of elliptic and interface problems on complex, non-smooth, or curved meshes.
• To achieve optimal convergence rates and high-order accuracy for these problems.

Main Methods:

• Construction of new basis functions on curved sides/faces of curvilinear elements using polynomial traces.
• Analysis of a modified weak Galerkin method incorporating these basis functions.
• Application to elliptic equations and interface problems on curvilinear polytopal meshes with Lipschitz continuous edges/faces.

Main Results:

• The proposed basis functions are linearly independent and suitable for curved domains.
• Optimal convergence rates for L2 and H1 errors are proven for the modified weak Galerkin method.
• The method demonstrates capability for handling complex boundaries and achieving arbitrary high-order accuracy.

Conclusions:

• The novel basis functions significantly improve the weak Galerkin method's applicability to complex geometries.
• The developed numerical approach provides an effective tool for problems with less smooth boundaries or interfaces.
• Theoretical findings are validated through numerical algorithm discussion and supporting test cases.