Quadratic Interpolation & 2nd Order Diff. Equations

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Quadratic interpolation can be used in the weak form of a second order differential equation, but it is not strictly necessary. The weak form requires both the weighting function and the primary variable to have non-zero first order derivatives to define the stiffness matrix, which can be achieved with linear approximations if the problem permits. However, linear elements lack interelement continuity, which may lead to issues with secondary variable continuity. In contrast, quadratic elements ensure continuity of secondary variables. Therefore, while quadratic interpolation is beneficial for maintaining continuity, it is not mandatory for deriving the weak form.
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For a second order differential equation, is it necessary to derive a weak form if quadratic interpolation are used?
 
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For the weak form of a second order differential equation, both weighting function(w) & the primary variable(u) are required to have non zero first order derivatives(in order to define the stiffness matrix), ie linear approximation can suffice provided the problem allows. Linear elements will not provide interelement continuity though, hence continuity of secondary variables cannot be guaranteed . Quadratic elements provide SV continuity.
 

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