A Dual-Mixed Approximation for a Huber Regularization of Generalized $p$-Stokes Viscoplastic Flow Problems

Abstract

In this paper, we extend a dual-mixed formulation for a nonlinear generalized Stokes problem to a Huber regularization of the Herschel-Bulkey flow problem. The present approach is based on a two-fold saddle point nonlinear operator equation for the corresponding weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of Newton differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments to investigate the behavior and efficiency of the method.

A Discontinuous Galerkin and Semismooth Newton Approach for the Numerical Solution of the Nonhomogeneous Bingham Flow

“…Summarizing, we can conclude that system (32) is well-posed (see [21] for further details). Moreover, by following a similar analysis as the one in [13] we can state that the SSN iteration converges superlinearly locally.…”

“…Hence, the right-hand side is nonnegative on a sphere of radius r := C r /C R . Consequently, by Theorem 3.4, there exists a solution to the fixed-point problem Φ(ρ n+1 h , u n+1 h , p n+1 h , z n+1 h ) = 0, where the fixed-point map (31) is the solution operator for the fully discrete problem (21).…”

“…Theorem 3.3 (Existence of discrete solutions). Problem (21) with initial data (ρ 1 h , u 1 h ) and (ρ 0 h , u 0 h ) admits at least one solution…”

“…The use of this differentiation concept is justified, since it is well known that both the max function and the Frobenius norm are slantly differentiable in finite dimension spaces (see [13,20,21]). Further, the SSN approach has proved to be efficient and provided a linearization scheme which is superlinear convergent when applied to discretized viscoplastic models, as discussed in the referred contributions.…”

“…Given the discussion above, the SSN linearization for system (21), about (ρ n+1 h , u n+1 h , p n+1 h , z n+1 h ), gives the following problem: find…”