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Superconvergence in the      L 2 {L^2}      -norm for the Galerkin approximation of the integral equation      L u = f Lu = f       is studied, where  L  is a strongly elliptic pseudodifferential operator on a smooth, closed or open curve. Let      u h {u_h}       be the Galerkin approximation to  u . By using the  K -operator, an operator that averages the values of      u h {u_h}      , we will construct a better approximation than      u h {u_h}       itself. That better approximation is a legacy of the highest order of convergence in negative norms. For Symm’s equation on a slit the same order of convergence can be recovered if the mesh is suitably graded.

The 𝐾-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates