AbstractPenlty coupling techniques on an interface boundary, artificial or material, are first presented for combining the Ritz–Galerkin and finite element methods. An optimal convergence rate first is proved in the Sobolev norms. Moreover, a significant coupling strategy,L+ 1 =O(|lnh|), between these two methods are derived for the Laplace equation with singularities, whereL+ 1 is the total number of particular solutions used in the Ritz–Galerkin method, andhis the maximal boundary length of quasiuniform elements used in the linear finite element method. Numreical experiments have been carried out for solving the benchmark model: Motz's problem. Both theoretical analysis and numreical experiments clearly display the importance of penalty‐combined methods is solving elliptic equations with singular