We　consider　a　family　of　elliptic　periodic　boundary　value　problems.　Our　aim　is　to　solve　the　elliptic　problems　numerically.　To　reach　this　goal,　we　construct　biorthogonal　periodic　wavelets　for　the　periodic　Sobolev　space　and　apply　them　to　solve　the　elliptic　periodic　problems　in　which　wavelet-Galerkin　method　is　used.　Numerical　experiments　indicate　that　our　method　is　very　efficient.