Abstract: Brinkman-Forchheimer equations (BF equations) describe the motion of the incompressible fluid under the strong nonlinearities. The accurate treatment of the incompressibility condition is critical for the numerical treatment of the BF equations. The penalty treatment is introduced to relax the incompressibility condition. In order to obtain the well-posedness  of the penalty problem, the pressure term is eliminated by using the penalty term, and an equivalence between the monotonous nonlinear elliptical problem and a minimization problem of corresponding energy functional is proposed. From the LBB condition, the existence and uniqueness of the variational problem are obtained. The convergence with respected to the penalty parameter is proved. Finally, the existence and uniqueness of the finite dimensional approximating problem are derived, and the error estimate based on the conforming finite element discretization is obtained. Numerical results show that the penalty finite element approximation is effective.

Key words: Brinkman-Forchheimer equation, penalty method, conforming finite element method, error estimate