Gage's Theorem

Gage's Theorem is a fundamental result in numerical analysis that addresses the accuracy of finite element methods used to approximate solutions to differential equations. It states that the error in the approximation depends on the size of the elements used in the mesh; smaller elements lead to more accurate solutions. Additionally, the theorem emphasizes that the convergence rate (how quickly the approximation improves as the mesh is refined) depends on the smoothness of the true solution and the degree of the polynomial used in the approximation. Overall, Gage's Theorem provides a mathematical foundation for predicting and improving the precision of numerical simulations.