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Given PDE: $\nabla^2 u(x,y)=(α^2-β^2 ) u(x,y)$

The domain is the square $[0,1] \times [0,1]$, which the boundary nodes are $(0,0),(1,0),(1,1),(0,1)$.

It is given that $u(0.5,0.5)=1$.

We would like to know how to started with using linear approximation to get the equation which connected those nodes?

Updates: If we would like to solve the PDE before linear approximation, what should we do?

Hiven
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  • This is way too broad. There is a lot that goes into solving PDEs with FEM. The translation into weak form, choosing basis functions, mesh algorithms etc. You need to clarify what specifically you need help with. – Eff May 04 '18 at 06:52
  • It seems that you have only one finite element. Is that true? And the additional condition is not an initial condition, because there is no time involved. And how are the boundary conditions at the edges of the square? – Han de Bruijn May 05 '18 at 08:51
  • Dear Eff, Thanks for the question, I would like to ask how to start using the linear approximation to get the equations which connect the nodes? – Hiven May 05 '18 at 16:25
  • Dear Han de Bruijn, Thanks for the remind. Sorry for the misleading. There are 4 triangular elements in total. 1:[(0,0)(1,0)(.5,.5)]2:[(.5,.5)(1,0)(1,1)]3:[(.5,.5)(1,1,)(0,1)]4:[(0,0)(.5,.5)(0,1)]...The boundary condition of the square of u(x,y)=0 – Hiven May 05 '18 at 17:23

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