PDEs
Given a Parabolic PDE, one wishes to construct a weak solution by projecting a solution of the original PDE onto finite-dimensional subspaces of
- Take an orthonormal basis
of by normalizing the eigenfunctions of (see Eigenvalues and Eigenfunctions of Elliptic Operators) - Define the approximants:
where the coefficients satisfy - Solve the linear system of Ordinary Differential Equations:
to find the unique Absolutely Continuous Function . - Take
and use energy estimates to show convergence to a unique weak solution: