This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Ampère equation on general triangular grids. This is done by establishing an equivalent (in the viscosity sense) Hamilton-Jacobi-Bellman formulation of the Monge-Ampère equation. A significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built-in property of the new formulation. Moreover, this new approach allows one to tap the wealthy numerical methods, such as semi-Lagrangian schemes, for Hamilton-Jacobi-Bellman equations to solve Monge-Ampère type equations. It is proved that the considered numerical methods are monotone, pointwise consistent and uniformly stable. Consequently, its solutions converge uniformly to the unique convex viscosity solution of the Monge-Ampère Dirichlet problem. A superlinearly convergent Howard's algorithm, which is a Newton-type method, is utilized as the nonlinear solver to take advantage of the monotonicity of the scheme. Numerical experiments are also presented to gauge the performance of the proposed numerical method and the nonlinear solver.

In this note we study the convergence of monotone P1 finite element methods on unstructured meshes for fully nonlinear Hamilton-Jacobi-Bellman equations arising from stochastic optimal control problems with possibly degenerate, isotropic diffusions. Using elliptic projection operators we treat discretisations which violate the consistency conditions of the framework by Barles and Souganidis. We obtain strong uniform convergence of the numerical solutions and, under non-degeneracy assumptions, strong L 2 convergence of the gradients. I IntroductionHamilton-Jacobi-Bellman (HJB) equations, which are of the formwhere the L α are linear first-or second order operators and d α ∈ L 2 , characterise the value function of optimal control problems. Indeed, one possibility to introduce the notion of solution of (1) is via the underlying optimal control structure. An alternative approach is to use the monotonicity properties of the operator which leads to the concept of viscosity solutions. While these perceptions are essentially equivalent [17, p.72] both views have been instructive for the design and analysis of numerical methods.The former approach, based on the discretisation of the optimal control problem before employing the Dynamic Programming Principle, has been proposed in the setting of finite elements in [26,7,8], see also the review article [22] and the references therein. Regarding finite difference methods we refer to the book [23]. The latter approach, which is also adopted in this note, was firmly established with the contribution [3] by Barles and Souganidis in 1991, providing an abstract framework for the convergence to viscosity solutions. Starting with [20,21] techniques were developed to quantify the rate of convergence; more recent works are [1,13]. A third direction was opened by the method of vanishing moments which neither enforces discrete maximum principles nor makes use of the underlying optimal control structure but relies on a higher order regularisation [16]. For a more comprehensive review of the state-of-the-art in the numerical solution of fully non-linear second order equations we refer to [15].In the traditional finite element analysis the multiplicative testing with hat functions is viewed as the discrete analogue of the multiplicative testing procedure to define weak solutions of the (variational) differential equation. While elements of this viewpoint are implicitly used in Section VII on gradient convergence, we would like to stress a second interpretation: multiplication with hat functions as regularisation of the residual. Consider for a moment the linear problem −a(x)∆u(x) = f (x) with smooth functions a and u as well as a hat function φ at the node y ℓ . Let P be the orthogonal projection onto the approximation space with respect to the scalar product 〈v, w〉 = ∇v · ∇w dx (given suitable boundary

We present the first direct experimental evidence for the existence of magnetic surface polaritons in a uniaxial antiferromagnet.The modes are excited in FeF2 by attenuated total reflection with the uniaxis along an applied magnetic field Ho. The resonance is observed as an attenuation of the reflected intensity as the frequency of the far-infrared beam is scanned. We find nonreciprocity between the +Ho and -Ho spectra as a consequence of the nonreciprocity of the surface polariton modes when Ho is introduced. We also observe broad dips due to the excitation of bulk modes. PACS numbers: 75.50.Ee, 75.30.Pd, 76.50.+g, 78.20.Ci Surface polaritons are now well established as a sensitive probe in surface analysis. The study of surface plasmon polaritons, for example, provides valuable information about material parameters associated with surface plasma oscillations such as the effective masses of charge carriers and surface charge densities. This technique has been applied to studies of metals [1] and semiconductors and semiconductor superlattices [2 -5].In ferromagnets, the study of magnetostatic surface spin waves by Brillouin light scattering has yielded important information on interface exchange constants, bulk and surface anisotropies, surface magnetization, and spinreorientation transitions for thin films and superlattices [6 -9]. In contrast, the experimental study of surface modes and the surface structure of antiferromagnets has been almost completely neglected. This is despite the fact the magnetic surface structure in antiferromagnets has recently been emphasized in a number of fundamental works [10], and also has technological importance in the exchange biasing of ferromagnetic films [11].We report the first direct experimental observation of magnetic surface polaritons on the uniaxial antiferromagnet FeF2 by attenuated total reIIection (ATR). The present development of ATR for magnetic surface excitations opens up the possibility of a wide range of experimental studies of magnetic surfaces in the far-infrared (FIR) spectral range. This technique should be applicable to studying surface modes and parameters for a wide variety of other magnetic materials. These include ferrimagnets, easy-plane antiferromagnets [12], rare earth magnets with helical orderings [13,14], and rare earth and antiferromagnetic superlattices [15].Magnetic surface polaritons (MSPs) were theoretically discussed first for ferromagnets [16] and later for antiferromagnets [17,18]. Subsequently a large theoretical literature has appeared [15] but no direct measurements of MSP have yet been reported. The experiments are challenging because it is necessary to establish ATR with high-frequency resolution at low temperatures.One of the key features to emerge from the theoretical studies is that MSP propagation is nonreciprocal, i.e. , reversing the direction of propagation (or reversing the applied field) changes the ATR rellectively. In our experimental results we indeed find a pronounced nonreciprocal reAectively. In addition, when th...

In this article we analyse the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous Galerkin method under minimal regularity assumptions. The main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem. In order to deal with the non-conformity of the method and to avoid overpenalisation of jumps across interelement boundaries, the careful construction of a reflexive subspace of the space of bounded variation, which compactly embeds into L 2 (Ω), and of a lifting operator, which is compatible with the nonlinear diffusion coefficient, are required. An equivalent skew-symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method. Numerical experiments underline the robustness of the proposed algorithm.

A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617-637, 2005) and Carstensen and Hu (J Numer Math 107(3):473-502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied to the Laplace, Stokes, and Lamé equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature. The estimate recovers known results as well as provides new bounds to a number of schemes.