Using Finite Elements Method on a PDE With No Solution

Applies Finite Element Method to a PDE which has no solution.

A simple partial differential equation (PDE) with boundary conditions is examined:

d/dx( x dy/dx ) = x
y(0) = y(1) = 0.

Integrate the PDE twice to get its solution. Then apply the boundary conditions and get a contradiction. The boundary value problem(BVP)has no solution.

Regardless, apply the finite elements method (FEM) using piecewise linear basis functions. The FEM is successfully completed without a hitch. There is no sign that the problem is insolvable.

This occurs because the FEM makes assumptions which restrict the solution space. Within this restricted space the BVP does have a solution.

The point is that one cannot blindly rely on a numerical technique to produce the correct answer to a problem. Numerical methods need to be supplemented with analysis.