Partial differential equations (PDE's) are one of the most fundamental applications of mathematics. They describe phenomena of the physical, natural and social science such as fluids, gravitational and electromagnetic fields, and the human body. They also play an important role in fields like aircraft simulation, computer graphics, and weather prediction.
There exist many partial differential equations, but from the view point of mathematics, three important examples are Laplace's equation, diffusion equation, and wave equation. The general linear 2 dimensional equation of the PDE's is can be written as follows
In this paper we will perform benchmarks on the Laplace's equation and diffusion equation.
Elliptic equation like Laplace equation is commonly found in multidimensional steady state problems. We can write the simplest form of elliptic equation (Poisson equation) by simplifying the equation (6.1) as follows