The general motivation for using a quadrature method is based on its algorithmic simplicity. We shall present several variations of quadrature methods depending on the order of the given operator. When applying these methods all the presented formulations lead to matrix forms which are easy to implement. On the other hand, many of the quadrature methods have a restricted convergence rate even in case of smooth data. But there are also some good exceptions; for equations of the second kind, singular integral equations and hypersingular equations we shall find cases where the convergence rate is of any polynomial order, and even exponential. This chapter is mainly based on the articles [SS93],[SS94],[SS95] and [Sar91] but includes also some new results.
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