How to solve $$ \begin{cases} y'(x)=y(x)\\ y(0)=1 \end{cases} $$ using finite element method?? I mean Bubnov-Galerkin approximation method along with Ritz minimization. I've try for some times and it looks like i don't understand something.
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This is an initial value problem. In order to solve numerically using FEM methods, you need another boundary condition, like $y(A) = B$. Then the problem becomes a "boundary condition" problem, which is slightly different. – the_candyman Jun 12 '17 at 20:40
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So the bottom line is: don't solve time-stepping with FEM; use a finite difference scheme instead. Believe me: I've tried both. – Han de Bruijn Apr 21 '18 at 20:24
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To solve ivp you need a last basis function that is free. This gives a singular stiffness matrix if there is a 2nd order derivative, I have not yet tried for the case given. Note that for a variable coefficient linear ode like $y'(x)+a(x)y(x)=0$ there are two possibilities of discretization : $\sum a_ky_k\phi_k(x)$ or $\sum a_ky_l\phi_k(x)\phi_l(x)$. In the latter case there are terms like $a_{k+1}y_k$ appearing – QuantumPotatoïd Jan 08 '22 at 21:28