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We consider two distinct cases: (i)<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is a relatively open subset of<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma _\\infty $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>\u0393<\/mml:mi><mml:mi>\u221e<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with fractal boundary (e.g. the interior of the Koch snowflake in the case<jats:inline-formula><jats:alternatives><jats:tex-math>$$n=3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>); (ii)<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is a compact fractal subset of<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma _\\infty $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>\u0393<\/mml:mi><mml:mi>\u221e<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with empty interior (e.g. the Sierpinski triangle in the case<jats:inline-formula><jats:alternatives><jats:tex-math>$$n=3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>). In both cases our numerical simulation strategy involves approximating the fractal screen<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>by a sequence of smoother \u201cprefractal\u201d screens, for which we compute the scattered field using boundary element methods that discretise the associated first kind boundary integral equations. We prove sufficient conditions on the mesh sizes guaranteeing convergence to the limiting fractal solution, using the framework of Mosco convergence. 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