Discontinuous Viscosity Research Articles

A mini immersed finite element method for two-phase Stokes problems on Cartesian meshes

Abstract This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.

On anisotropic parabolic equation with nonstandard growth order

ABSTRACT In this paper, the existence and the uniqueness of an evolutionary anisotropic p i ( x ) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic p ( x ) − Laplace equation with convection term via L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc.

Non-overlapping domain decomposition algorithms with only primal velocity unknowns for the discontinuous viscosity Stokes problem

In this paper, non-overlapping domain decomposition algorithms for the Stokes problem with a discontinuous viscosity coefficient are proposed and analyzed. A pair of inf–sup stable finite element spaces with discontinuous pressure finite element functions is employed to obtain a discrete Stokes system. By introducing a non-overlapping subdomain partition for the given finite element spaces, the discrete Stokes system is reformulated into an algebraic system on dual unknowns, where the continuity on the velocity unknowns at the subdomain vertices is enforced strongly and the continuity on the remaining interface velocity unknowns is enforced weakly using the dual unknowns. The resulting algebraic system on the dual unknowns is then solved by an iterative method. The coarse problem in the resulting algebraic system is only related to the strongly coupled primal velocity unknowns and it is thus obtained as a symmetric positive definite matrix equation. To accelerate the iteration convergence, practical versions of lumped and Dirichlet preconditioners are proposed and their condition numbers are also analyzed for the resulting dual algebraic system. Numerical results are also included to confirm our condition number estimates.

Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids

A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integrals indirectly through a Cartesian grid-based method. Since the jump conditions of the single-fluid Stokes problems can be easily decoupled, the modified MAC scheme is adopted here and the existing fast solver can be applicable for the resulting linear saddle system. The computed numerical solutions are second order accurate in discrete ℓ2-norm for velocity and pressure as well as the gradient of velocity, and also second order accurate in maximum norm for both velocity and its gradient, even in the case of high contrast viscosity coefficient, which is demonstrated in numerical tests. In addition, moving interface driven by the surface tension is also provided to validate the efficiency of the KFBI method.

A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces

A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces

A face‐centred finite volume method for high‐contrast Stokes interface problems

AbstractA new face‐centred finite volume method (FCFV) for Stokes problems involving sharp interfaces is proposed. Two formulations, based on two strong forms of the Stokes problem, and using different mixed variables, are presented. Particular attention is paid to the symmetry of the resulting system of global equations, and a simple rewriting of the interface boundary condition is proposed to ensure that one of the formulations preserves the symmetry of the linear system that is usually lost when considering material interfaces. Four numerical examples are considered to test the implementation numerically by performing mesh convergence studies, in two and three dimensions. The examples account for discontinuous viscosity as well as the effect of surface tension. The results show that one of the formulations is less sensitive to the numerical stabilisation used in FCFV methods but does not preserve the symmetry of the global system, whereas the other formulation is more sensitive to the stabilisation, but preserves the symmetry of the resulting system of equations. The FCFV method appears as a promising alternative for the simulation of viscous flow involving internal boundaries on conformal meshes. The potential application of the FCFV method for the purpose of geodynamic modelling is discussed.

Backward reachability approach to state-constrained stochastic optimal control problem for jump-diffusion models

AbstractIn this paper, we consider the stochastic optimal control problem for jump-diffusion models with state constraints. In general, the value function of such problems is the discontinuous viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation since the regularity cannot be guaranteed at the boundary of the state constraint. By adapting the stochastic target theory, we obtain an equivalent representation of the original value function by means of the backward reachable set. We then show that this backward reachable can be characterized by the zero-level set of the auxiliary value function for the unconstrained stochastic control problem, which includes two additional unbounded controls as a consequence of the martingale representation theorem. We prove that the auxiliary value function is the unique continuous viscosity solution of the associated HJB equation, which is the second-order nonlinear partial integro-differential equation (PIDE). Our paper provides an explicit way to characterize the original (possibly discontinuous) value function as a zero-level set of the continuous solution of the auxiliary HJB equation. The proof of the existence and uniqueness requires a new technique due to the unbounded control sets and the presence of the singularity of the corresponding Lévy measure in the nonlocal operator of the HJB equation.

A matrix-free multilevel preconditioner for the generalized Stokes problem with discontinuous viscosity

In this paper we present a matrix-free multilevel solver for the generalized Stokes problem with discontinuous viscosity. The algorithm is based on the FGMRES iteration preconditioned with a block-smoothed multigrid method. Numerical experiments on unstructured grids indicate that for a broad range of smoother parameters, the convergence speed is only weakly dependent on both the number of unknowns and the jumps of the viscosity coefficient.

Viscosity solutions for doubly nonlinear evolution equations

We extend the theory of viscosity solutions to treat scalar-valued doubly nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e. we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g. we prove the existence of a solution to a rate-independent level-set mean curvature flow. Finally, we discuss in detail a rate-independent ordinary differential equation stemming from a problem with non-convex energy. We show that the solution obtained by maximal minimizing movements and the solution obtained by the vanishing viscosity method coincide with the upper and lower Perron solutions and show the emergence of a rate-independent hysteresis loop.

О сходимости разностных схем метода опорных операторов для ротационных операций векторного анализа на тетраэдрических сетках

One approach to the description of the metric properties of an irregular grid is proposed for discretizing repeated rotational operations of vector analysis (rot rot) as applied to modeling the magnetic and electric fields. On the basis of the support operator method and the used grid metric, integrally consistent vector analysis operations rot, div, grad are constructed that are necessary to obtain convergence estimates for difference schemes for repeated rotational operations, as well as solving physical problems with discontinuous magnetic viscosity, dielectric permeability or thermal resistance of a medium. On smooth solutions of the model magnetostatic problem on a tetrahedral grid with first order of accuracy in the mean-square sense, the convergence of the constructed difference schemes with zero eigenvalue of the spectral problem is proved. At the same time, no restrictions are imposed on an irregular tetrahedral mesh, except for its non-degeneracy.

Discontinuous solutions of $U_t + H(U_x) = 0$ versus measure-valued solutions of $u_t + [H(u)]_x = 0$

Let H be a bounded Lipschitz continuous function. We discuss some recent results concerning discontinuous viscosity solutions of the Hamilton–Jacobi equation U_t + H(U_x) = 0 0, signed Radon measure-valued entropy solutions of the conservation law u_t + [H(u)]_x = 0 , and their connection.

Discontinuous viscosity solutions of first-order Hamilton–Jacobi equations

We study the Cauchy problem for the simplest first-order Hamilton–Jacobi equation in one space dimension, with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. Uniqueness of discontinuous viscosity solutions is proven, if the initial data function has a finite number of jump discontinuities. Main ingredients of the proof are the barrier effect of spatial discontinuities of a solution (which is linked to the boundedness of the Hamiltonian), and a comparison theorem for semicontinuous viscosity subsolution and supersolution. These are defined in the spirit of the paper [H. Ishii, Perron’s method for Hamilton–Jacobi equations, Duke Math. J. 55 (1987) 368–384], yet using essential limits to introduce semicontinuous envelopes. The definition is shown to be compatible with Perron’s method for existence and is crucial in the uniqueness proof. We also describe some properties of the time evolution of spatial jump discontinuities of the solution, and obtain several results about singular Neumann problems which arise in connection with the above referred barrier effect.

A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

A second order accuracy preserving method for moving contact lines with Stokes flow

The immersed interface method (IIM) has been widely used in simulations of multiphase flows with closed interfaces. We generalize IIM to simulate the moving contact line problems, which are modeled by the Stokes equation with the Navier-slip boundary condition and the contact angle condition. With the help of variational formulation, the contact angle condition can be combined with the interfacial kinematics in a weak form. A parametric finite element method (parametric FEM) is applied to solve for the interface motion as well as the curvature, which are in turn used to update the correction terms for the irregular points in IIM. The hybrid IIM-parametric FEM method is Cartesian grid based, and achieves second order accuracy not only in the velocity field but also in the interface and the contact line motion. This is validated by numerical results. Moreover, we generalize the method to account for discontinuous viscosity. Various numerical experiments are presented in the study of droplet motion and contact angle hysteresis.

Global existence to a diagonal hyperbolic system for any BV initial data

In this paper, we study the existence of solutions for a diagonal hyperbolic system, that is not necessarily strictly hyperbolic, in one space dimension, considering discontinuous BV initial data without any restrictions on the size of its norm. This system appears naturally in various physical domains, particularly in isentropic gas dynamics and dislocation dynamics in materials. In the case of strictly hyperbolic systems, an existence and uniqueness of a discontinuous solution result is available for BV initial data with small norm, whereas several existence and uniqueness results have been presented for non-decreasing continuous solutions. In the present paper, we show the global in time existence of discontinuous viscosity solutions to a diagonal hyperbolic system for every initial data of bounded total variation, without the assumption that the system is strictly hyperbolic. Up to our knowledge, this is the first global existence result of large discontinuous solutions to this system.

Discontinuous Solutions of Hamilton–Jacobi Equations Versus Radon Measure-Valued Solutions of Scalar Conservation Laws: Disappearance of Singularities

Let H be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton–Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton–Jacobi equation and signed singular measures in case of the conservation law.

$ BV $ solution for a non-linear Hamilton-Jacobi system

<p style='text-indent:20px;'>In this work, we are dealing with a non-linear eikonal system in one dimensional space that describes the evolution of interfaces moving with non-signed strongly coupled velocities. For such kind of systems, previous results on the existence and uniqueness are available for quasi-monotone systems and other special systems in Lipschitz continuous space. It is worth mentioning that our system includes, in particular, the case of non-decreasing solution where some existence and uniqueness results arose for strictly hyperbolic systems with a small total variation. In the present paper, we consider initial data with unnecessarily small <inline-formula><tex-math id="M2">\begin{document}$ BV $\end{document}</tex-math></inline-formula> seminorm, and we use some <inline-formula><tex-math id="M3">\begin{document}$ BV $\end{document}</tex-math></inline-formula> bounds to prove a global-in-time existence result of this system in the framework of discontinuous viscosity solution.

Least squares spectral method for the two-dimensional Stokes interface problems

This paper deals with the pseudo-spectral least squares method to approximate the solution of the Stokes interface problem, that is, the Stokes equations with discontinuous viscosity and singular source term. The Stokes interface equations are adjusted in each sub-domain along with the interface conditions. Then the equations are reformulated as a first order system by introducing stress tensor as a new independent variable. The continuous and discrete norm least squares functionals using Legendre and Chebyshev weights for the first order system are defined and shown to be equivalent to appropriate product norms. The spectral convergence of the proposed method is presented. Numerical examples are also given to demonstrate the method and analysis.

Utility maximisation in a factor model with constant and proportional transaction costs

We study the problem of maximising expected utility of terminal wealth under constant and proportional transaction costs in a multidimensional market with prices driven by a factor process. We show that the value function is the unique viscosity solution of the associated quasi-variational inequalities and construct optimal strategies. While the value function turns out to be truly discontinuous, we are able to establish a comparison principle for discontinuous viscosity solutions which is strong enough to argue that the value function is unique, globally upper semicontinuous, and continuous if restricted to either borrowing or non-borrowing portfolios.

A direct IIM approach for two-phase Stokes equations with discontinuous viscosity on staggered grids

In this paper, a direct immersed interface method (IIM) is proposed to solve two-phase incompressible Stokes equations with an interface and a piecewise constant viscosity on staggered grids. The velocity components and the pressure are placed in different grid points and the Marker and Cell (MAC) scheme is used for discretizing the momentum and continuity equations at regular grid points. At irregular grid points, correction terms are added to the finite difference scheme to offset the discontinuities. The correction terms are determined directly by an interpolation scheme using the values of both the velocity and pressure at nearby grid points. The resulted linear system of equations is rank-one deficient and is solved by the Uzawa iterative method. In each Uzawa iteration, an inner GMRES solver is used and preconditioned by the discrete Laplacian. The computed numerical solutions are second order accurate in the L∞ norm for both the velocity and pressure, which is demonstrated in numerical tests. Compared with the augmented interface method (AIIM), one of advantages of this approach is that it avoids the costs for introducing augmented variables and difficulties in solving them from the corresponding Schur complement system. Hence, this new method is easier to implement and computationally more efficient.