The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressiblefluid exhibiting phase transitions between a liquid and a vapor phase in the presence ofcapillarity effects close to phase boundaries. Standard numerical discretizations areknown to violate discrete versions of inherent energy inequalities, thus leading tospurious dynamics of computed solutions close to static equilibria (e.g.,parasitic currents). In this work, we propose a time-implicit discretization of theproblem, and use piecewise linear (or bilinear), globally continuous finite element spacesfor both, velocity and density fields, and two regularizing terms where correspondingparameters tend to zero as the mesh-size h > 0 tends to zero.Solvability, non-negativity of computed densities, as well as conservation of mass, and adiscrete energy law to control dynamics are shown. Computational experiments are providedto study interesting regimes of coefficients for viscosity and capillarity.

We propose and analyse two convergent fully discrete schemes to solve the incompressible Navier-Stokes-Nernst-Planck-Poisson system. The first scheme converges to weak solutions satisfying an energy and an entropy dissipation law. The second scheme uses Chorin's projection method to obtain an efficient approximation that converges to strong solutions at optimal rates.

The incompressible MHD equations couple Navier-Stokes equations with Maxwell's equationsto describe the flow of a viscous, incompressible, and electrically conducting fluid ina Lipschitz domain $\Omega \subset \mathbb{R}^3$ .We verify convergence of iterates of different coupling anddecoupling fully discrete schemes towards weak solutions forvanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouplesthe computation of velocity field, pressure, and magnetic fields atevery iteration step.

We consider a degenerate parabolic system which modelsthe evolution of nematic liquid crystal with variable degree of orientation.The systemis a slight modificationto that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical fully discrete finite element method for this regularized system, and we establish the (subsequence) convergence of this finite element approximation to the solution of the regularized system as the mesh parameters tend to zero; andto a solution of the original degenerate parabolic system when the the mesh and regularization parameters all approach zero. Finally, numerical experiments are included which show the formation, annihilation and evolution of line singularities/defects in such models.

The magnetization of a ferromagnetic sample solves anon-convex variational problem, where its relaxation by convexifyingthe energy density resolves relevantmacroscopic information. The numerical analysis of the relaxed modelhas to deal with a constrained convexbut degenerated, nonlocal energy functional in mixed formulation formagnetic potential u and magnetization m.In [C. Carstensen and A. Prohl, Numer. Math.90(2001) 65–99], the conforming P1 - (P0)d -element in d=2,3 spatialdimensions is shown to lead toan ill-posed discrete problem in relaxed micromagnetism, and suboptimalconvergence.This observation motivated anon-conforming finite element method which leads toa well-posed discrete problem, with solutions converging atoptimal rate.In this work, we provide both an a priori and a posteriori error analysis for twostabilized conforming methods which account for inter-element jumps of thepiecewise constant magnetization.Both methods converge at optimal rate;the new approach is applied to a macroscopic nonstationary ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ].

This paper studies the gradient flow of a regularized Mumford-Shah functionalproposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L2 x L∞ initial data possesses a global weak solution, and it has a unique global in timestrong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H1 x H1 ∩ L∞ . A family of fully discrete approximation schemes using low order finite elements is proposed for the gradient flow. Convergence of a subsequence (resp. the whole sequence)of the numerical solutions to a weak solution (resp. the strong solution) of the gradient flow is established as the mesh sizes tend to zero, and optimal and suboptimal order error estimates, which depend on $\frac{1}{{\varepsilon}}$ and $\frac{1}{k_{\varepsilon}}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^{\frac12})$ . Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow,and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship betweenthe regularized gradient flow (characterized by a small positive parameterε, see (1.7)) and the minimal surface flow [21]and the prescribed mean curvature flow [16].Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence tothe regularized gradient flow problem as h,k → 0, and to the total variation gradient flow problem as h,k,ε → 0in general cases.Provided that the regularized gradient flow problem possessesstrong solutions, which is proved possible if the datum functionsare regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the meshrelation k = O(h2). In particular, it is shown thatall error bounds depend on $\frac{1}{\varepsilon}$ onlyin some lower polynomial order for small ε.

The minimization of nonconvex functionals naturally arises inmaterials sciences where deformation gradients in certain alloys exhibitmicrostructures. For example, minimizing sequences of the nonconvexEricksen-James energy can be associated with deformations in martensitic materials thatare observed in experiments[2,3]. — From the numericalpoint of view, classical conforming and nonconforming finite elementdiscretizations have been observed to give minimizerswith their quality being highlydependent on the underlying triangulation, see [8,24,26,27] for a survey. Recently, a newapproach has been proposed and analyzed in [15,16]that is based on discontinuous finite elements to reduce the pollution effectof a general triangulation on the computed minimizer.The goal of the present paper isto propose and analyzean adaptive method,giving a more accurate resolution of laminated microstructure on arbitrary grids.