Search

We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn–Hilliard–Navier–Stokes (CHNS) system, combined with logarithmic Flory–Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard ℓ ∞ (0 , T ; ℓ 2) ∩ ℓ 2 (0 , T ; H h 2) error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the ℓ ∞ (0 , T ; H h 1) ∩ ℓ 2 (0 , T ; H h 3) error analysis for the phase variable and the ℓ ∞ (0 , T ; ℓ 2) analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result. To our knowledge, it will be the first work to establish an optimal rate convergence estimate for the Cahn–Hilliard–Navier–Stokes system with a singular energy potential. [ABSTRACT FROM AUTHOR]

We present a second order accurate numerical scheme for a nonlinear hyperbolic equation with an exponential nonlinear term. The solution to such an equation is proven to have a conservative nonlinear energy. Due to the special nature of the nonlinear term, the positivity is proven to be preserved under a periodic boundary condition for the solution. For the numerical scheme, a highly nonlinear fractional term is involved, for the theoretical justification of the energy stability. We propose a linear iteration algorithm to solve this nonlinear numerical scheme. A theoretical analysis shows a contraction mapping property of such a linear iteration under a trivial constraint for the time step. We also provide a detailed convergence analysis for the second order scheme, in the ℓ ∞ ( 0 , T ; ℓ ∞ ) norm. Such an error estimate in the maximum norm can be obtained by performing a higher order consistency analysis using asymptotic expansions for the numerical solution. As a result, instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O ( Δ t 3 + h 4 ) convergence in ℓ ∞ ( 0 , T ; ℓ 2 ) norm, which leads to the necessary ℓ ∞ error estimate using the inverse inequality, under a standard constraint Δ t ≤ C h . A numerical accuracy check is given and some numerical simulation results are also presented. [ABSTRACT FROM AUTHOR]

Abstract: An conforming mixed finite element method has been proposed for the coupled Darcy–Stokes flow in Kanschat and Rivière (2010) [12], which imposes normal continuity on the velocity field strongly across the Darcy–Stokes interface. Here, we develop an a posteriori error estimator for this conforming mixed method, and prove its global reliability and efficiency. Due to the strong coupling on the interface, special techniques need to be employed in the proof. This is the main difference between this paper and Babuška and Gatica’s (2010) work [38], in which they analyzed an a posteriori error estimator for the mixed formulation using weakly coupled interface conditions. [Copyright &y& Elsevier]

In this paper, an energy stable, second-order mixed finite element scheme is proposed and analyzed for the thin film epitaxial growth model with slope selection. We employ second-order backward differentiation formula (BDF) scheme with a second-order stabilized term, which guarantees the long time energy stability to approximate the continuous model. In terms of the convergence analysis, the key difficulty to derive an optimal rate spatial estimate is associated with the appearance of the gradient operator in the nonlinear terms, which may lead to a loss of optimal accuracy order. To overcome this well-known difficulty, we make use of some auxiliary techniques over triangular elements, and obtain an optimal convergence rate O (h q + 1 + Δ t 2) , in comparison with O (h q + Δ t 2) rate from a standard projection estimate. Furthermore, we use an efficient preconditioned steepest descent (PSD) solver for the numerical implementation. A few numerical examples are presented to validate the stability and convergence. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to numerical analysis of history-dependent variational– hemivariational inequalities arising in contact problems for viscoelastic material. We introduce both temporally semi-discrete approximation and fully discrete approximation for the problem, where the temporal integration is approximated by a trapezoidal rule and the spatial variable is approximated by the finite element method. We analyze the discrete schemes and derive error bounds. The results are applied for the numerical solution of a quasistatic contact problem. For the linear finite element method, we prove that the error estimation for the numerical solution is of optimal order under appropriate solution regularity assumptions. [ABSTRACT FROM AUTHOR]