Discrete Variational Inequality Research Articles

Hp-FEM for the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Mosolov problem: a priori and a posteriori error estimates

An hp-finite element discretization for the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Mosolov problem, a scalar variant of the Bingham flow problem but with the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Laplacian operator, is being analyzed. Its weak formulation is either a variational inequality of second kind or equivalently a non-smooth but convex minimization problem. For any α∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,\\infty )$$\\end{document} we prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the FE-solution of the corresponding discrete variational inequality. Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any “approximation” of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. We prove that any quasi-minimizer of those families of a posteriori error estimators satisfies an efficiency estimate. All our results contain known results for the Mosolov problem by setting α=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =2$$\\end{document}. Numerical results underline our theoretical findings.

The mixed method with two Lagrange multiplier formulations for the Signorini problem

Introducing stress as a new unknown, we consider the Signorini problem in the mixed form. The well-posedness theory of the continuous and discrete mixed variational inequalities has been established by reformulating them into projection problems. Two Lagrange multiplier formulations are introduced utilizing different projections. The equivalency among the Lagrange multiplier formulations and the mixed variational inequality is demonstrated in continuous and discrete sense, respectively. For the discrete mixed variational inequality, we develop the error estimates. Based on two Lagrange multiplier formulations, we design two Active/Inactive set algorithms and investigate the convergence. Several numerical experiments are conducted to verify the theoretical convergence rates of the finite element discretization and the iteration algorithms.

Dynamic user optimal route choice problem on a signalized transportation network

In this paper, we present a new route-based discrete time variational inequality (VI) ideal dynamic user optimal route choice (DUO) model on a transportation network with fixed signal timing at signalized intersections and a relaxation with gradient projection algorithm to solve the model. The new trait of our study is the queue length in red time and the throughput in green time for each cycle at each intersection can be found when traffic is in DUO status. This is different from other studies, in which either only average queue length or average delay in a period composing of multiple cycles can be found or traffic is not in DUO status. Furthermore, the optimal signal timings can be found by the algorithm. Under optimal signal timing, the total queuing delay of the network reaches minimum when traffic is in DUO status.

Additive Schwarz preconditioners for C0 interior penalty methods for the obstacle problem of clamped Kirchhoff plates

AbstractThe discrete variational inequalities resulting from interior penalty methods for the obstacle problem of clamped Kirchhoff plates can be solved by the primal‐dual active set algorithm. We develop and analyze additive Schwarz preconditioners for the auxiliary systems that appear in each iteration of the primal‐dual active set algorithm. Numerical results corroborate our theoretical estimates.

Error estimate for indirect spectral approximation of optimal control problem governed by fractional diffusion equation with variable diffusivity coefficient

In this paper an indirect spectral method of an optimal control problem governed by a space-fractional diffusion equation with variable diffusivity coefficient is studied. First-order optimality conditions of the proposed model are derived and the regularity of the solutions is analyzed. Indirect spectral methods via weighted Jacobi polynomials are built up based on the “first optimize, then discretize” strategy, and a priori error estimates of the discrete optimal control problem in weighted norms are derived. As proposed indirect spectral discrete schemes are designed to accommodate the impact of the variable coefficients, which are complicated and not formulated under the variational framework, conventional error estimate techniques of the discrete space-fractional optimal control problems do not apply. Novel treatments of the discrete variational inequality are developed to resolve the aforementioned issues and to support the error estimates. Numerical examples are presented to verify the theoretical findings.

Numerical Simulation Three-Dimensional Nonlinear Seepage in a Pumped-Storage Power Station: Case Study

Due to high water pressure in the concrete reinforced hydraulic tunnels, surrounding rocks are confronted with nonlinear seepage problem in the pumped storage power station. In this study, to conduct nonlinear seepage numerical simulation, a nonlinear seepage numerical model combining the Forchheimer nonlinear flow theory, the discrete variational inequality formulation of Signorini’s type and an adaptive penalized Heaviside function is established. This numerical seepage model is employed to the seepage analysis of the hydraulic tunnel surrounding rocks in the Yangjiang pumped-storage power station, which is the highest water pressure tunnel under construction in China. Moreover, the permeability of the surrounding rocks under high water pressure is determined by high pressure packer test and its approximate analytical model. It is shown that the flow in the surrounding rocks is particularly prone to become nonlinear as a result of the high flow velocities and hydraulic gradients in the nearby of the seepage-control measures and the high permeability fault. The nonlinear flow theory generates smaller flow rate than the Darcy flow theory. With the increase of nonlinear flow, this observation would become more remarkable.

Additive Schwarz preconditioners for the obstacle problem of clamped Kirchhoff plates

When the obstacle problem of clamped Kirchhoff plates is discretized by a partition of unity method, the resulting discrete variational inequalities can be solved by a primal-dual active set algorithm. In this paper we develop and analyze additive Schwarz preconditioners for the systems that appear in each iteration of the primal-dual active set algorithm. Numerical results that corroborate the theoretical estimates are also presented.

Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation

This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate and fourth order non degenerate parabolic equations, in this work the a priori estimates of the error between the discrete solution and the weak solution to which it converges are obtained for a degenerate fourth order parabolic equation. The theoretical error estimates obtained in the present case state that the norms of the approximation errors, calculated on the support of the solution in the proper functional spaces, are bounded by power laws of the discretization parameters with exponent 1/2, while in the case of the classical Cahn-Hilliard equation with constant mobility the exponent is 1. The estimates are finally succesfully validated by simulation results in one and two space dimensions.

A Cahn‐Hilliard–type equation with application to tumor growth dynamics

We consider a Cahn‐Hilliard–type equation with degenerate mobility and single‐well potential of Lennard‐Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn‐Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.

Stabilized finite element methods for a blood flow model of arteriosclerosis

In this article, a blood flow model of arteriosclerosis, which is governed by the incompressible Navier–Stokes equations with nonlinear slip boundary conditions, is constructed and analyzed. By means of suitable numerical integration approximation for the nonlinear boundary term in this model, a discrete variational inequality for the model based on stabilized finite elements is proposed. Optimal order error estimates are obtained. Finally, numerical examples are shown to demonstrate the validity of the theoretical analysis and the efficiency of the presented methods. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2063–2079, 2015

Optimal Convergence for Discrete Variational Inequalities Modelling Signorini Contact in 2D and 3D without Additional Assumptions on the Unknown Contact Set

The basic $H^1$-finite element error estimate of order $h$ with only $H^2$-regularity on the solution has not been yet established for the simplest two-dimensional Signorini problem approximated by a discrete variational inequality (or the equivalent mixed method) and linear finite elements. To obtain an optimal error bound in this basic case and also when considering more general cases (e.g., the three-dimensional problem, quadratic finite elements), additional assumptions on the exact solution (in particular on the unknown contact set, see [Z. Belhachmi and F. Ben Belgacem, Math. Comp., 72 (2003), pp. 83--104; S. Hueber and B. Wohlmuth, SIAM J. Numer. Anal., 43 (2005), pp. 156--173; B. Wohlmuth, A. Popp, M. Gee, and W. Wall, Comput. Mech. 49 (2012), pp. 735--747] had to be used. In this paper, we consider finite element approximations of the two-dimensional and three-dimensional Signorini problems with linear and quadratic finite elements. In the analysis, we remove all the additional assumptions and pr...

An Approach to Nonlinear Viscoelasticity via Metric Gradient Flows

We formulate quasistatic nonlinear finite-strain viscoelasticity of rate type as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure time-dependent frame indifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the time-incremental problem. Because of the missing compactness the limit of vanishing timesteps can be obtained only by proving some kind of strong convergence. We show that this is possible in the one-dimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the time-continuous limit in a specific case.

Full Discretisations for Nonlinear Evolutionary Inequalities Based on Stiffly Accurate Runge–Kutta and hp-Finite Element Methods

The convergence of full discretisations by implicit Runge–Kutta and nonconforming Galerkin methods applied to nonlinear evolutionary inequalities is studied. The scope of applications includes differential inclusions governed by a nonlinear operator that is monotone and fulfills a certain growth condition. A basic assumption on the considered class of stiffly accurate Runge–Kutta time discretisations is a stability criterion which is in particular satisfied by the Radau IIA and Lobatto IIIC methods. In order to allow nonconforming hp-finite element approximations of unilateral constraints, set convergence of convex subsets in the sense of Glowinski–Mosco–Stummel is utilised. An appropriate formulation of the fully discrete variational inequality is deduced on the basis of a characteristic example of use, a Signorini-type initial-boundary value problem. Under hypotheses close to the existence theory of nonlinear first-order evolutionary equations and inequalities involving a monotone main part, a convergence result for the piecewise constant in time interpolant is established.

Boundary element methods for variational inequalities

In this paper we present a priori error estimates for the Galerkin solution of variational inequalities which are formulated in fractional Sobolev trace spaces, i.e. in $$\widetilde{H}^{1/2}(\Gamma )$$ H ? 1 / 2 ( Γ ) . In addition to error estimates in the energy norm we also provide, by applying the Aubin---Nitsche trick for variational inequalities, error estimates in lower order Sobolev spaces including $$L_2(\Gamma )$$ L 2 ( Γ ) . The resulting discrete variational inequality is solved by using a semi-smooth Newton method, which is equivalent to an active set strategy. A numerical example is given which confirms the theoretical results.

Finite Element Method for Stokes Equations under Leak Boundary Condition of Friction Type

Stokes equations are considered with the leak boundary condition of friction type. We propose a discrete variational inequality based on one of the P1/P1-stabilized, P1b/P1, and P2/P1 finite elements. We show that a suitable numerical-integration approximation for the nonlinear term preserves in a discrete sense some properties known in continuous problems. Those properties are essential to justify our error estimate and numerical implementation in a consistent way. Numerical examples are presented to confirm the theoretical results.

A Primal-Dual Finite Element Approximation for a Nonlocal Model in Plasticity

We study the numerical approximation of a static infinitesimal plasticity model of kinematic hardening with a nonlocal extension. Here, the free energy to be minimized is a combination of the elastic energy and an additional term depending on the curl of the plastic variable. First, we introduce the stress as dual variable and provide an equivalent primal-dual formulation resulting in a local flow rule. The discretization is based on curl-conforming Nédélec elements. To obtain optimal a priori estimates, the finite element spaces have to satisfy a uniform inf-sup condition. This can be guaranteed by adding locally defined face and element bubbles. Second, the discrete variational inequality system is reformulated as a nonlinear equality. We show that the classical radial return algorithm applied to the mixed inequality formulation is equivalent to a semismooth Newton method for the nonlinear system of equations. Numerical results illustrate the convergence of the applied discretization and the solver.

Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method

The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading to a discrete variational inequality of saddle point type in each time step. In each iteration of the primal-dual active set method a linearized system resulting from the discretization of two coupled elliptic equations which are defined on different sets has to be solved. We show local convergence of the primal-dual active set method and demonstrate its efficiency with several numerical simulations.

Numerical solution procedures for the morning commute problem

This paper discusses solution techniques for the morning commute problem that is formulated as a discrete variational inequality (VI). Various heuristics have been proposed to solve this problem, mostly because the analytical properties of the path travel time function have not yet been well understood. Two groups of “non-heuristic” algorithms for general VIs, namely projection-type algorithms and ascent direction algorithms, were examined. In particular, a new ascent direction method is introduced and implemented with a heuristic line search procedure. The performance of these algorithms are compared on simple instances of the morning commute problem. The implications of numerical results are discussed.

A numerical solution to seepage problems with complex drainage systems

Seepage problems with complex drainage systems are commonly encountered in civil engineering, with strong non-linearity. A numerical solution based on the Finite Element Method combining the substructure technique with a variational inequality formulation of Signorini’s type is proposed to solve these problems. The aims of this work are to accurately characterize the boundary conditions of the drainage systems, to reduce the difficulty in mesh generation resulting from the drainage holes with small radius and dense spacing, and to eliminate the singularity at the seepage points and the resultant mesh dependency. Numerical stability and robustness of the proposed method are guaranteed by an adaptive procedure for progressively relaxing the penalized Heaviside function associated with the formulation of the discrete variational inequality. Two challenging numerical examples are presented to validate the effectiveness and robustness of the proposed method.

Solving the Dynamic User Optimal Assignment Problem Considering Queue Spillback

This paper studies the dynamic user optimal (DUO) traffic assignment problem considering simultaneous route and departure time choice. The DUO problem is formulated as a discrete variational inequality (DVI), with an embeded LWR-consistent mesoscopic dynamic network loading (DNL) model to encapsulate traffic dynamics. The presented DNL model is capable of capturing realistic traffic phenomena such as queue spillback. Various VI solution algorithms, particularly those based on feasible directions and a line search, are applied to solve the formulated DUO problem. Two examples are constructed to check equilibrium solutions obtained from numerical algorithms, to compare the performance of the algorithms, and to study the impacts of traffic interacts across multiple links on equilibrium solutions.