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The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal\u00a0<jats:italic>h<\/jats:italic>-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. 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