Finite Element Method

• Differential Equation are continuous systems
• Computers can't solve continuous system. One idea is convert it to a discrete the problem. e.g the derivative of a function is the slope at a single point, however in discrete, the derivative is the slope of line joining two points.
• Another different approach to this is the idea Galerkin had. He had the idea of trial functions. Close enough functions to approximate the actual solutions.
• At that time he used 2 or 3 trial functions, but nowadays, we have computers so, we use 1000s of simple functions. This way the problem of solving the diffrential equation is changed to finding the coefficients with which to multiply the trial functions.

FEM was initially developed for Structural Problems. And in those problems the motion is very small. A bridge deflects ever so slightly. But on the other hand in fluid dynamics, a river flows, and thus it's a completely different problem. Much difficult.

• For Solid Mechanics, we are good.
• For Fluid mechanics, we have much work to do.
• For Gas mechanics, we have much much work to do.

Also Wavelets! People think why don't we use wavelets as trial functions?