Spline qualocation methods for variable-coefficient elliptic equations on curves

Here the stability and convergence results of oqualocation methods providin g additional orders of convergence are extended from the special class of p seudodifferential equations with constant coefficient symbols to general cl assical pseudodifferential equations of strongly and of oddly elliptic type . The analysis exploits localization in the form of frozen coefficients, ps eudohomogeneous asymptotic symbol representation of classical pseudodiffere ntial operators, a decisive commutator property of the qualocation projecti on and requires qualocation rules which provide sufficiently many additiona l degrees of precision for the numerical integration of smooth remainders. Numerical examples show the predicted high orders of convergence. Mathemati cs Subject Classification (1991): 65R20.