We generalize the accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration methods for large sparse saddle-point problems. These methods involve four iteration parameters whose special choices can recover the preconditioned HSS and accelerated HSS iteration methods. Also a new efficient case is introduced and we theoretically prove that this new method converges to the unique solution of the saddle-point problem. Numerical experiments are used to further examine the effectiveness and robustness of iterations.

In this paper, we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in [Kang & March, IEEE T. Image Process., 16 (2007), 2251-2261]. We use fast algorithms to solve the subproblems, such as the dual projection methods, primal-dual methods and augmented Lagrangian methods. With a special penalization term, some special algorithms are presented. Numerical experiments are given to demonstrate the performance of the proposed methods. We successfully show that our algorithms are effective and efficient due to two reasons: the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately (even 2 inner iterations of the subproblem are enough). It is also observed that better PSNR values are produced using the new algorithms.

We consider the lowest-order Raviart–Thomas mixed finite elementmethod for second-order elliptic problems on simplicial meshes intwo and three space dimensions. This method produces saddle-pointproblems for scalar and flux unknowns. We show how to easily andlocally eliminate the flux unknowns, which implies the equivalencebetween this method and a particular multi-point finite volumescheme, without any approximate numerical integration. The matrixof the final linear system is sparse, positive definite for alarge class of problems, but in general nonsymmetric. We next showthat these ideas also apply to mixed and upwind-mixed finiteelement discretizations of nonlinear parabolicconvection–diffusion–reaction problems. Besides the theoreticalrelationship between the two methods, the results allow forimportant computational savings in the mixed finite elementmethod, which we finally illustrate on a set of numericalexperiments.