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 $L_2$-norm error analysis of the $HP$-version with numerical integration Bull. Korean Math. Soc. 2002 Vol. 39, No. 1, 9-22 Published online March 1, 2002 Ik-Sung Kim Korea Maritime University Abstract : We consider the $hp-$version to solve non-constant coefficients elliptic equations with Dirichlet boundary conditions on a bounded, convex polygonal domain $\Omega$ in $R^2$. To compute the integrals in the variational formulation of the discrete problem we need the numerical quadrature rule scheme. In this paper we consider a family $G_p =\{ I_m \}$ of numerical quadrature rules satisfying certain properties. When the numerical quadrature rules $I_m \in G_p$ are used for calculating the integrals in the stiffness matrix of the variational form we will give its variational form and derive an error estimate of ${\|u-\widetilde u_p^h \|}_{0,\Omega}$. Keywords : the $hp$ version, numerical quadrature rules, non-constant coefficients elliptic equations MSC numbers : 65G99 Full-Text :