By　boundary　element　method,　we　present　a　numerical　iterative　process　for　solving　a　free　third　boundary　problem　modeling　tumor　growth　with　spectral　accuracy.　The　piecewise　quadratic　curves　are　fitted　to　maintain　local　smoothness　of　the　boundary　at　every　node.　The　double-layer　and　single-layer　potentials　with　weakly　singular　kernels　are　evaluated　with　spectral　accuracy.　The　method　of　characteristics　is　employed　to　transform　interfacial　velocity　PDE　into　discrete　ODEs.　The　numerical　integral　formula　for　weakly　singular　operator　with　logarithmic　singularity　is　deduced　and　the　convergence　and　error　are　presented.　The　nonradially　symmetric　solutions　of　the　free　boundary　problem　on　a　perturbed　boundary　are　provided　to　test　the　accuracy　and　effectiveness　of　the　numerical　method.　(C)　2018　Elsevier　B.V.　All　rights　reserved.