Abstract. This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number k, optimal order (with respect to h) error estimate in the broken H 1 -norm and sub-optimal order estimate in the L 2 -norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime (i.e., k 2 h 1). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to k) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [23,24,35].

We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u t − u + ε −2 f (u) = 0 arising from phase transition in materials science, where ε is a small parameter known as an "interaction length". The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ε. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u 0 . In particular, all our error bounds depend on 1 ε only in some lower polynomial order for small ε. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18,19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow.

This paper concerns with numerical approximations of solutions of second order fully nonlinear partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for second order fully nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called vanishing moment method, hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods with "guaranteed" convergence. The main idea of the proposed vanishing moment method is to approximate a second order fully nonlinear PDE by a higher order, in particular, a fourth order quasilinear PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.

No abstract

Summary.We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation u t + (ε u − ε −1 f (u)) = 0, where ε > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on 1 ε only in some lower polynomial order for small ε. The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as ε → 0 in [29].

We consider complex-valued acoustic and elastic Helmholtz equations with the first order absorbing boundary condition in a star-shaped domain in N for N ≥ 2. It is known that the elliptic regularity coefficients depend on the frequency ω, and have singularities for both zero and infinite frequency. In this paper, we obtain sharp estimates for the coefficients with respect to large frequencies. It is proved that the elliptic regularity coefficients are bounded by first or second order polynomials in ω for large ω. The crux of our analysis is to establish and make use of Rellich identities for the solutions to the acoustic and elastic Helmholtz equations. Our results improve the earlier estimates of Refs. 10 and 11, which were carried out based on layer potential representations of the solutions of the Helmholtz equations.

No abstract

Abstract. This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampère equation det(D 2 u 0 ) = f (> 0) based on the vanishing moment method which was proposed recently by the authors in [19]. In this approach, the second order fully nonlinear Monge-Ampère equation is approximated by the fourth order quasilinear equation −ε∆ 2 u ε + det D 2 u ε = f . It was proved in [17] that the solution u ε converges to the unique convex viscosity solution u 0 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 u ε of the regularized fourth order problem, which computes simultaneously u ε and the moment tensor σ ε := D 2 u ε . Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter ε, for the errors u ε − u ε h and σ ε − σ ε h . Finally, we present a detailed numerical study on the rates of convergence in terms of powers of ε for the error u 0 − u ε h and σ ε − σ ε h , and numerically examine what is the "best" mesh size h in relation to ε 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.