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fa.practical evaluation – Continuity of Weak Solution of Elliptic PDE Answer

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fa.practical evaluation – Continuity of Weak Solution of Elliptic PDE

I’m investigating the next benchmark elliptic PDE with combined Dirichlet-Neumann border situation:

$-Delta u=f$ on $Omega$; $u=0$ on $Gamma_D$; $left<n,nabla u=gright>$ on $Gamma_N$

Computationally, we might discover the feeble resolution utilizing finite-dimensional check duty house $V$ by requiring:

$int_Omega left<nabla u,nabla vright> dOmega = int_Omega fvdOmega+ int_{Gamma_N}gvdsquadforall vin V$

I’m questioning whether or not the feeble resolution $u$ is steady on the duty house. In different phrases, if there’s one other duty house $V’$ and I’ve a practical $d$ that’s the metric of duty house: $d(V,V’)$, would the answer $u$ breathe steady on $d$? As a sensible instance in engineering, deem FEM mesh discreting the PDE on a mesh, if I perturb a vertex of the mesh, would $u$ change constantly?

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