An upwind finite element method for solving hyperbolic equations

In this paper, we establish and comprehensively examine a stabilized weak Galerkin (WG) finite element framework specifically engineered for multidimensional hyperbolic systems. The developed methodology employs an upwind stabilization mechanism, which proves essential for maintaining numerical stability while achieving optimal convergence characteristics, particularly when implementing piecewise constant approximations. We establish and rigorously demonstrate optimal error bounds within both H¹ and L² norms, thereby constructing a mathematically sound theoretical foundation. We execute extensive computational investigations to validate the theoretical developments and conclusively establish the effectiveness and robustness of the proposed methodology.