Weak Galerkin Finite Element Methods

Research in weak Galerkin finite element methods focuses on developing and analyzing numerical techniques for solving differential equations. This area explores novel discretization schemes, particularly those that relax continuity constraints on finite element spaces. Investigations include the use of polytopal meshes, which allow for complex geometries, and the application of these methods to challenging problems such as the Stokes equations used in fluid dynamics. The theoretical underpinnings involve rigorous error analysis and the establishment of convergence properties for various types of differential equations.

The development of efficient and accurate numerical solvers has broad applicability across Arkansas industries. For example, understanding fluid flow is critical for the state's agricultural sector, particularly in irrigation and water management. Similarly, the simulation of physical processes is essential for materials science and engineering applications relevant to manufacturing. Accurate modeling can also inform public health initiatives by providing tools for epidemiological modeling and the analysis of disease spread.

This research intersects with fields including differential equations analysis, finite element analysis, and materials science. The work benefits from and contributes to advancements in computational mathematics, with potential for collaboration across institutions within Arkansas and beyond.