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A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form O(hp + τk+1) in the L∞(0, T; H¹)-norm is established, where h and τ denote the spatial and temporal mesh sizes, respectively, and (p, k) is the degree of the space-time finite elements. Numerical experiments are provided to validate the theoretical results on the convergence rates and conservation properties. The effectiveness of the proposed methods in preserving the shape of a soliton wave is also demonstrated by numerical results. [ABSTRACT FROM AUTHOR]

This paper presents a new narrow-stencil finite difference method for approximating viscosity solutions of Hamilton--Jacobi--Bellman equations. The proposed finite difference scheme naturally extends the Lax--Friedrichs scheme for first order fully nonlinear PDEs to second order fully nonlinear PDEs which are approximated by Lax--Friedrichs-like numerical operators. The crux for constructing such a numerical operator is to introduce a stabilization term, which is called a "numerical moment" and corresponds to the numerical viscosity term in the original Lax--Friedrichs scheme for first order PDEs. It is proved that the proposed Lax--Friedrichs-like scheme has a unique solution and is stable in both the ℓ²-norm and the ℓ∞-norm. Moreover, the convergence of the proposed finite difference scheme to the viscosity solution of the underlying Hamilton--Jacobi--Bellman equation is also established using a novel discrete comparison argument. [ABSTRACT FROM AUTHOR]

This paper studies finite element approximations of the stochastic Allen--Cahn equation with gradient-type multiplicative noise that is white in time and correlated in space. The sharp interface limit--as the diffuse interface thickness vanishes--of the stochastic Allen--Cahn equation is formally a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the deterministic mean curvature flow. Two fully discrete finite element methods which are based on different time-stepping strategies for the nonlinear term are proposed. Strong convergence with sharp rates for both fully discrete finite element methods is proved. This is done with the crucial help of the Holder continuity in time with respect to the spatial L2-norm and H¹-seminorm for the strong solution of the stochastic Allen--Cahn equation. It also relies on the fact that high moments of the strong solution are bounded in various spatial and temp oral norms. Numerical experiments are provided to gauge the performance of the proposed fully discrete finite element methods and to study the interplay of the geometric evolution and gradient-type noise. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty DG methods for spatial discretization. They differ from each other on how the nonlinear term is treated; one of them is based on fully implicit time-stepping and the other uses energy-splitting time-stepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on ϵ-1 only in some low polynomial orders, instead of exponential orders. Similar to [X. Feng and A. Prohl, Numer. Math., 74 (2004), pp. 47-84], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space HM1(Ω) and it is larger than the finite element space. This difficulty is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved by Feng and Prohl. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods. [ABSTRACT FROM AUTHOR]

This paper develops and analyzes an interior penalty discontinuous Galerkin (IPDG) method using piecewise linear polynomials for the indefinite time harmonic Maxwell equations with the impedance boundary condition in the three-dimensional space. The main novelties of the proposed IPDG method include the following: first, the method penalizes not only the jumps of the tangential component of the electric field across the element faces but also the jumps of the tangential component of its vorticity field; second, the penalty parameters are taken as complex numbers of negative imaginary parts. For the differential problem, we prove that the sesquilinear form associated with the Maxwell problem satisfies a generalized weak stability (i.e., inf-sup condition) for star-shaped domains. Such a generalized weak stability readily infers wave-number explicit a priori estimates for the solution of the Maxwell problem, which plays an important role in the error analysis for the IPDG method. For the proposed IPDG method, we show that the discrete sesquilinear form satisfies a coercivity for all positive mesh size h, wave number k, and for general domains including nonstar-shaped ones. In turn, the coercivity estimate easily yields the well-posedness and stability estimates (i.e., a priori estimates) for the discrete problem without imposing any mesh constraint. Based on these discrete stability estimates, by adapting a nonstandard error estimate technique of [X. Feng and H. Wu, SIAM J. Numer. Anal., 47 (2009), pp. 2872-2896, X. Feng and H. Wu, Math. Comp., 80 (2011), pp. 1997-2024], we derive both the energy-norm and the L²-norm error estimates for the IPDG method in all mesh parameter regimes including preasymptotic regime (i.e., k²h ≳ 1). Numerical experiments are also presented to gauge the theoretical results and to numerically examine the pollution effect (with respect to k) in the error bounds. [ABSTRACT FROM AUTHOR]

This article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations (PDEs), a relatively new subarea within numerical PDEs. Due to their ever increasing importance in mathematics itself (e.g., differential geometry and PDEs) and in many scientific and engineering fields (e.g., astrophysics, geostrophic fluid dynamics, grid generation, image processing, optimal transport, meteorology, mathematical finance, and optimal control), numerical solutions to fully nonlinear second order PDEs have garnered a great deal of interest from the numerical PDE and scientific communities. Significant progress has been made for this class of problems in the past few years, but many problems still remain open. This article intends to introduce these current advancements and new results to the SIAM community and generate more interest in numerical methods for fully nonlinear PDEs. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes some fully discrete (multiphysics) finite element methods for a displacement-pressure model which describes swelling dynamics of polymer gels under mechanical constraints. By introducing an "elastic pressure" we first present a reformulation of the original model. We then propose a time-stepping scheme which decouples the PDE system at each time step into two subproblems, one of which is a Stokes-like problem for the displacement vector field (of the solid network of the gel) and the other is a diffusion problem for a "pseudopressure" field (of the solvent of the gel). To make such a multiphysics approach feasible, it is vital to find admissible constraints to resolve the uniqueness issue for the Stokes-like problem and to construct a good boundary condition for the diffusion equation so that it also becomes uniquely solvable. The key to the first difficulty is to discover certain conserved quantities for the PDE solution, and the solution to the second difficulty is to use the Stokes-like problem to generate a boundary condition for the diffusion problem. The time-stepping scheme allows one to use any convergent Stokes solver (and its code) together with any convergent diffusion equation solver (and its code) to solve the polymer gel model. In the paper, the Taylor-Hood mixed finite element method combined with the continuous linear finite element method are chosen as an example to present the ideas and to demonstrate the viability of the proposed multiphysics approach. It is proved that, under a mesh constraint, both the semidiscrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for both semidiscrete and fully discrete methods. Numerical experiments are also presented to show the performance of the proposed approach and methods. [ABSTRACT FROM AUTHOR]

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampère equation det(D²u0) = f (> 0) based on the vanishing moment method which was proposed recently by the authors in [X. Feng and M. Neilan, J. Scient. Comp., DOI 10.1007/s10915-008-9221-9, 2008]. In this approach, the second-order fully nonlinear Monge-Amp`ere equation is approximated by the fourth order quasilinear equation -ϵΔ²uϵ + detD²uϵ = f. It was proved in [X. Feng, Trans. AMS, submitted] that the solution ue converges to the unique convex viscosity solution u0 of the Dirichlet problem for the Monge-Ampère equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second-order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann-Miyoshi-type mixed finite element methods for approximating the solution uϵ of the regularized fourth-order problem, which computes simultaneously ue and the moment tensor σϵ := D²uϵ . Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter ϵ, for the errors uϵ - uϵ h and σ0 - σϵ h . Finally, we present a detailed numerical study on the rates of convergence in terms of powers of e for the error u0 -uϵ h and σϵ -σϵh , and numerically examine what is the "best" mesh size h in relation to e in order to achieve these rates. Due to the strong nonlinearity of the underlying equation, the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem. To overcome the difficulty, we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations. [ABSTRACT FROM AUTHOR]

Motivated by emerging applications from imaging processing, this paper studies the heat flow of a generalized p-harmonic map into spheres for the whole spectrum, 1 ≤ p < ∞, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a BV -solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms A · ∇v and A Λ ∇v, which pair a divergence-L¹, or a divergence-measure, tensor field A and a BV -vector field v. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the p-harmonic map heat flow, and the convergence of the proposed numerical method is also established. [ABSTRACT FROM AUTHOR]

The Ericksen-Leslie model describes dynamics of low molar-mass nematic liquid crystals, where the spatiotemporal distribution of defects defining texture is represented by the director unit vector field d : ΩT → S2. It consists of the Navier-Stokes equations with an extra viscous stress tensor, and a convective harmonic map heat flow equation to govern the dynamics of the director field. Two fully discrete finite element methods, for a regularized system using the Ginzburg-Landau regularization and for the limiting Ericksen-Leslie model, are proposed, and well-posedness and related discrete energy laws are established. For the regularized model, unconditional convergence of finite element solutions towards weak solutions of the continuum model as well as convergence towards measure-valued solutions of the limiting Ericksen-Leslie model are verified when the mesh parameters and the regularization parameter successively tend to zero. Computational experiments are also presented to show the importance of balancing numerical and regularization parameters, to compare regularized and direct approaches, and to show numerically the finite-time formation, annihilation, and evolution of point defects. [ABSTRACT FROM AUTHOR]

Three fully discrete finite element methods are developed for approximating wave maps into the sphere based on two different approaches. The first method is an explicit scheme and the numerical solution satisfies the sphere-constraint exactly at every node. The second and third methods are implicit schemes which are based on a penalization approach, their numerical solutions satisfy the sphere-constraint approximately, and the quality of approximations is controlled by a small penalization parameter. Discrete energy conservation laws which mimic the underlying differential conservation law are established, and convergence of all proposed methods is proved. Computational experiments are also provided to validate the proposed methods and to present numerical evidence for possible finite-time blow-ups of the wave maps. [ABSTRACT FROM AUTHOR]

An explicit fully discrete finite element method, which satisfies the nonconvex side constraint at every node, is developed for approximating the p-harmonic flow for p ϵ (1, ∞). Convergence of the method is established under certain conditions on the domain and mesh. Computational examples are presented to demonstrate finite-time blow-ups and qualitative geometric changes of weak solutions of the p-harmonic flow. [ABSTRACT FROM AUTHOR]

This paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the Navier-Stokes equations and the Cahn-Hilliard equation, which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids. In the model the two sets of equations are coupled through an extra phase induced stress term in the Navier-Stokes equations and a fluid induced transport term in the Cahn-Hilliard equation. Fully discrete mixed finite element methods are proposed for approximating the coupled system, it is shown that the proposed numerical methods satisfy a mass conservation law, and a discrete energy law which is analogous to the basic energy law for the phase field model. The convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law. As a by-product, the convergence result also provides a constructive proof of the existence of weak solutions to the Navier-Stokes-Cahn-Hilliard phase field model. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach. [ABSTRACT FROM AUTHOR]

We present some two-level nonoverlapping and overlapping additive Schwarz methods for a discontinuous Galerkin method for solving second order elliptic problems. It is shown that the condition numbers of the preconditioned systems are of the order O(H/h) for the nonoverlapping Schwarz methods and of the order O(H/δ) for the overlapping Schwarz methods, where h and H stand for the fine-mesh size and the coarse-mesh size, respectively, and δ denotes the size of the overlaps between subdomains. Numerical experiments are provided to gauge the efficiency of the methods and to validate the theory. [ABSTRACT FROM AUTHOR]