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In the p-adaptive version of the finite element method, elements are allowed to have shape functions with arbitrary different polynomial orders. Therefore regarding a 2D problem with quadrilateral hierarchic shape functions, the polynomial orders of four edge-modes of an element, $p_e^1,p_e^2,p_e^3,p_e^4$, and its internal-mode, $p_i$, might be in general not the same.

In such a case, what degree of Gauss-Legendre quadrature should be used to numerically integrate the area integrals on an element? My guess is that $\mathrm{max}(p_e^1,p_e^2,p_e^3,p_e^4,p_i)$ determines what accuracy order of quadrature to use.

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Yes, a typical choice is to use the maximum -- which happens to be based on the polynomial degree of the finite element used for the element (notwithstanding the fact that some of its degrees of freedom might be constrained to enforce continuity with neighbors).

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