Nonlinear elliptic and parabolic equations of the second order

The mixed virtual element method on curved edges in two dimensions

We show that the discrete Sinkhorn algorithm - as applied in the setting of Optimal Transport on a compact manifold - converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited.

The Sinkhorn algorithm, parabolic optimal transport and geometric Monge-Amp\`ere equations

In "I. Smears, E. Suli, \emph{Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordes coefficients. SIAM J. Numer Anal., 51(4):2088-2106, 2013}" the authors designed and analysed a discontinuous Galerkin finite element method for the approximation of solutions to elliptic partial differential equations in nondivergence form. The results were proven, based on the assumption that the computational domain was convex and \emph{polytopal}. In this paper, we extend this framework, allowing for Lipschitz continuous domains with piecewise curved boundaries.

A DGFEM for nondivergence form elliptic equations with Cordes coefficients on curved domains

Adaptive C0 interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

In this paper we present and analyze a discontinuous Galerkin finite element method (DGFEM) for the approximation of solutions to elliptic partial differential equations in nondivergence form, with...

A Discontinuous Galerkin Finite Element Method fOR Uniformly Elliptic Two Dimensional Oblique Boundary-Value Problems

We provide a unified analysis of a posteriori and a priori error bounds for a broad class of discontinuous Galerkin and $C^0$-IP finite element approximations of fully nonlinear second-order elliptic Hamilton--Jacobi--Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions in $H^2$ of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for the a priori error analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda--Talenti inequality to piecewise polynomial vector fields.

/pdf/unified-analysis-of-discontinuous-galerkin-and-c-0-interior-224bx8b1fm.pdf

Unified analysis of discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations

We address the numerical solution via Galerkin type methods of the Monge-Ampere equation with transport boundary conditions arising in optimal mass transport, geometric optics and computational mesh or grid movement techniques This fully nonlinear elliptic problem admits a linearisation via a Newton-Raphson iteration, which leads to an oblique derivative boundary value problem for elliptic equations in nondivergence form We discretise these by employing the nonvariational finite element method, which lead to empirically observed optimal convergence rates, provided recovery techinques are used to approximate the gradient and the Hessian of the unknown functions We provide extensive numerical testing to illustrate the strengths of our approach and the potential applications in optics and mesh movement

Ellya L. Kawecki

Papers

A DGFEM for nondivergence form elliptic equations with Cordes coefficients on curved domains

A Discontinuous Galerkin Finite Element Method fOR Uniformly Elliptic Two Dimensional Oblique Boundary-Value Problems

Adaptive C0 interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

Unified analysis of discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations

A finite element method for the Monge-Ampère equation with transport boundary conditions