Abstract: A new algorithm-Point-by-Point Lagrange multiplier method (PLMM) is proposed in analysis of the contact and friction problem of the face of the center crack on cracked Brazilian disk specimens subjected to shear-compressive loading. To avoid the global stiffness inversion, the Lagrange multiplier is transformed to local coordinate system at first. Then the iterative solutions of normal and tangential multipliers are obtained by using Gauss-Seidel iteration method with Point-by-Point technique and more attention are paid to the constraint modification of the tangential multiplier. The point multipliers are transformed to global coordinate system to solve for displacements iteratively. The comparison between the proposed method and the ordinary contact algorithm shows that the PLMM, involving no search of the inverse global stiffness matrix, requires less computer storage and has high computational efficiency. The stress intensity factors (SIFs) of Brazilian disk central crack with contact and friction are calculated under the two typical cases using the proposed method, showing good agreement with the literatures. The effects of the load angle and the interface crack length on displacements, SIFs and contact region are investigated. Besides, the numerical results indicate that the friction has a significant influence on the SIFs and the SIFs may be overestimated without considering crack surface contact and friction.