Finite Element Methods with Artificial Diffusion for Hamilton-Jacobi-Bellman Equations

In this short note we investigate the numerical performance of the method of artificial diffusion for second-order fully nonlinear Hamilton-Jacobi-Bellman equations. The method was proposed in (M. Jensen and I. Smears, arxiv:1111.5423); where a framework of finite element methods for Hamilton-Jacobi-Bellman equations was studied theoretically. The numerical examples in this note study how the artificial diffusion is activated in regions of degeneracy, the effect of a locally selected diffusion parameter on the observed numerical dissipation and the solution of second-order fully nonlinear equations on irregular geometries.Comment: Enumath 2011, version 2 contains in addition convergence rate

On the Convergence of Finite Element Methods for Hamilton-Jacobi-Bellman Equations

In this note we study the convergence of monotone P1 finite element methods on unstructured meshes for fully non-linear 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 L2 convergence of the gradients.Comment: Keywords: Bellman equations, finite element methods, viscosity solutions, fully nonlinear operators; 18 pages, 1 figur

Stable Crank-Nicolson Discretisation for Incompressible Miscible Displacement Problems of Low Regularity

In this article we study the numerical approximation of incompressible miscible displacement problems with a linearised Crank-Nicolson time discretisation, combined with a mixed finite element and discontinuous Galerkin method. At the heart of the analysis is the proof of convergence under low regularity requirements. Numerical experiments demonstrate that the proposed method exhibits second-order convergence for smooth and robustness for rough problems.Comment: Enumath 200

Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is presented and analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. More specifically, a model problem consisting of a system of semilinear parabolic advection-diffusion-reaction partial differential equations in each compartment, equipped with respective initial and boundary conditions, is considered. Nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. An interior penalty dG method is presented for this problem and it is analysed in the space-discrete setting. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection-dominated regime

Density and trace for graph spaces of first-order linear operators

We define and analyse graph spaces of first-order linear differential operators. In particular we consider the density of the set of smooth functions and the construction of a trace operator

Numerical solution of the simple Monge–Ampère equation with nonconvex dirichlet data on non-convex domains

The existence of a unique numerical solution of the semi-Lagrangian method for the simple Monge-Ampere equation is known independently of the convexity of the domain or Dirichlet boundary data - when the Monge-Ampere equation is posed as a Bellman problem. However, the convergence to the viscosity solution has only been proved on strictly convex domains. In this paper, we provide numerical evidence that convergence of numerical solutions is observed more generally without convexity assumptions. We illustrate how in the limit multivalued functions may be approximated to satisfy the Dirichlet conditions on the boundary as well as local convexity in the interior of the domai

Application of the Inverse Almost Ideal Demand System to Welfare Analysis

This paper presents the theoretical properties of the Inverse Almost Ideal De-mand System and applies the system on time series data for cod, herring and plaice in Denmark (1986 to 2001). Furthermore, the shortcoming of the Inverse Almost Ideal Demand System when applied to welfare analysis is discussed. The properties of the demand system show that - since the demand system is a second-order approximation to the true system - it does not have global appli-cability for welfare measurement. It may, therefore, not satisfy the conditions for calculation of consumer surplus (negative slope and positive point of inter-section with the price-axis). The theoretical point is illustrated by an empirical example of the Danish fish market. Using a vector auto regressive model in er-ror correction form to overcome the problem of non-stationarity of data, the In-verse Almost Ideal Demand System is estimated. For cod the intercept is nega-tive and for herring and plaice the slope of the demand function is positive in the data interval investigated. Thus, the estimated demand system is not suitable for welfare analysis.Inverse Almost Ideal Demand System, Welfare analysis, Co-integration and Fish

Discontinuous Galerkin methods for Friedrichs systems with irregular solutions

This work is concerned with the numerical solution of Friedrichs systems by discontinuous Galerkin “nite element methods (DGFEMs). Friedrichs systems are boundary value problems with symmetric, positive, linear “rst-order partial differential operators and allow the uni“ed treatment of a wide range of elliptic, parabolic, hyperbolic and mixed-type equations. We do not assume that the exact solution of a Friedrichs system belongs to a Sobolev space, but only require that it is contained in the associated graph space, which amounts to differentiability in the characteristic direction. We show that the numerical approximations to the solution of a Friedrichs system by the DGFEM converge in the energy norm under hierarchicalh- and p- re“nement. We introduce a new compatibility condition for the boundary data, from which we can deduce, for instance, the validity of the integration-by-parts formula. Consequently, we can admit domains with corners and allow changes of the inertial type of the boundary, which corresponds in special cases to the componentwise transition from in- to out”ow boundaries. To establish the convergence result we consider in equal parts the theory of graph spaces, Friedrichs systems and DGFEMs. Based on the density of smooth functions in graph spaces over Lipschitz domains, we study trace and extension operators and also investigate the eigensystem associated with the differential operator. We pay particular attention to regularity properties of the traces, that limit the applicability of energy integral methods, which are the theoretical underpinning of Friedrichs systems. We provide a general framework for Friedrichs systems which incorporates a wide range of singular boundary conditions. Assuming the aforementioned compatibility condition we deduce well-posedness of admissible Friedrichs systems and the stability of the DGFEM. In a separate study we prove hp-optimality of least-squares stabilised DGFEMs