Nonconforming Least Squares Spectral Element Method for Parabolic Problems with Corner Singularities

Abstract

Differential equations play a pivotal role in explaining and modelling various natural phenomena and man made applications in science and engineering. They describe how quantities change w.r.t. one another and are fundamental to mathematics, physics, chemistry, engineering, biology, economics, and many other fields. From predicting the motion of planets and the flow of electricity, to modelling population growth or the spread of diseases, differential equations allow us to express complex real-world phenomena through mathematical relationships. They form the foundation of many scientific laws, such as Newton s laws of motion, Fourier s law of heat transfer, Maxwell s equations of electromagnetism and Einstein s field equations, among many others. By solving differential equations, scientists and engineers can predict future behaviour, optimize systems, and gain deeper insights into the dynamics of change over various parameters of interest. However, for differential equations to be of much use, it is imperative to solve for the various unknowns. Nevertheless, it has been long known that obtaining closed form solution to these equations is at best a herculean task if not impossible. Various techniques have been developed over the years to address the issue, yet one theme stands out with the advent of digital computing machinery Numerical Methods . The basic idea in any numerical approach is discretization which refers to the finite dimensional approximation of the solution to some reasonable accuracy with the additional constraint of consistency and stability.