Using Finite Elements Method on a PDE With 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.
Cite As
Mark Mueller (2024). Using Finite Elements Method on a PDE With No Solution (https://www.mathworks.com/matlabcentral/fileexchange/31482-using-finite-elements-method-on-a-pde-with-no-solution), MATLAB Central File Exchange. Retrieved .
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