153 THE ITERATED GALERKIN METHOD FOR INTEGRAl EQUATIONS OF THE SECOND KIND I.H. Sloan l. INTRODUC'riON Consider the integral equation of the second kind (l.l) y(t) f(t) + f k(t,s)y(s)do(s) , n where n is either a bounded domain in d with a locally Lipschitz boundary or the smooth d -dimensional boundary of a bounded domain in JRd+l, and dO(s) is the element of volume or surface area, as appropriate. Writing the equation as (L2) y f + Ky , we shall assume ·that for each p in l ,; p ,; oo K is a compact linear operator in L p f E L p and the corresponding homogeneous equation has no non--trivial solution in L p tha·t a (unique) solution yEL p It follows then from the Fredholm theorem exis-'cs for each f E L p The Galerkin method, in which an approxima·te solution yh is sought in a fini>ce-dimensional space s 11 C L 00 (see Section 2 for details) , is a well understood numerical method for the solution of (l.l). Here vJe are more concerned with the iterated variant of the Galerkin method, Le. wi'ch the approximation obtained by substituting the Galerkin approximation yh into 'che right-hand side of the in·tegral equation, giving