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In the framework of the Jacobi-weighted Besov spaces, we analyze the convergence of the h-p version of finite element solutions on quasi-uniform meshes and the lower and upper bounds of errors for elliptic problems on polygons. Both lower and upper bounds are proved to be optimal in h and p, which leads to the optimal convergence of the h-p version of the finite element method with quasi-uniform meshes for elliptic problems on polygons. The results proved for the h-p version include the h-version with quasi-uniform meshes and the p-version with quasi-uniform degrees as two special cases. © 2007 Society for Industrial and Applied Mathematics.