We introduce contact express, the exponential pressure method for contact mechanics. Contact express is based on a primal variable reformulation of the proximal Galerkin method for contact mechanics, first introduced in [1]. The method has an implicit and explicit form; both forms amount to solving a sequence of linear elasticity problems with an implicit or explicit Dirichlet-type boundary condition, respectively. We apply the method to Signorini’s problem, the classical model of a three-dimensional linearly elastic body resting on a rigid, frictionless surface, and provide an implementation using the finite element software MFEM [2]. Our numerical results affirm mesh-independent convergence of the method, in line with the theory in proximal Galerkin. For certain bodies and simulations, we combine contact express with a Nitsche-type method to enforce the Dirichlet boundary condition weakly. Our results show that contact express is a promising approach for solving contact problems. [1] Jørgen S. Dokken and Patrick E. Farrell and Brendan Keith and Ioannis P.A. Papadopoulos and Thomas M. Surowiec, The latent variable proximal point algorithm for variational problems with inequality constraints, Computer Methods in Applied Mechanics and Engineering, Vol. 445, pp. 118181, 2025 [2] R. Anderson and J. Andrej and A. Barker and J. Bramwell and J.-S. Camier and J. Cerveny and V. Dobrev and Y. Dudouit and A. Fisher and Tz. Kolev and W. Pazner and M. Stowell and V. Tomov and I. Akkerman and J. Dahm and D. Medina and S. Zampini, MFEM: A Modular Finite Element Methods Library, Computers & Mathematics with Applications, Vol. 81, pp. 42-74, 2021