Initial data for boosted Kerr black hole are constructed in an axially symmetric case. Momentum and Hamiltonian constraints are solved numerically using finite element method (FEM) algorithms. Both Bowen--York and puncture boundary conditions are adopted and appropriate results are compared. Past and future apparent horizons are also found numerically and the Penrose inequality is tested in detail.

PACS numbers: 04.20.Cr, 04.20.Dw, 04.70.Bw