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MIXED INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS FOR ONE-DIMENSIONAL fully nonlinear SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS

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Journal of Computational Mathematics 2014年第2期32卷 107-135页

作者： Xiaobing Feng Thomas Lewis Department of Mathematics The University of Tennessee Knoxville TN 37996 U.S.A. Department of Mathematics and Statistics The University of North Carolina at Greensboro NC 27402 U.S.A.

This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations （pdes）. In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin （IP-DG） methods for approximating viscosity solutions of these fully nonlinear pdes. In order to capture discontinuities of the second order derivative uxx of the solution u, three independent functions p1,p2 and p3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG frame- work, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative uxz. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear pdes using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator which is consistent with the differential operator and satisfies certain monotonicity （called g-monotonicity） properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG frame- work. The g-monotonicity gives the DG methods the ability to select the mathematically ＂correct＂ solution （i.e., the viscosity solution） among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage

关键词： fully nonlinear pdes Viscosity solutions Discontinuous Galerkin methods.

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Numerical Solution of the Incompressible Navier-Stokes Equation by a Deep Branching Algorithm

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Communications in Computational Physics 2023年第7期34卷 261-289页

作者： Jiang Yu Nguwi Guillaume Penent Nicolas Privault School of Physical and Mathematical Sciences Nanyang Technological UniversitySingapore 637371Singapore

We present an algorithm for the numerical solution of systems of fully nonlinear pdes using stochastic coded branching *** approach covers functional nonlinearities involving gradient terms of arbitrary orders,and it requires only a boundary condition over space at a given terminal time T instead of Dirichlet or Neumann boundary conditions at all times as in standard *** implementation relies on Monte Carlo estimation,and uses neural networks that perform a meshfree functional estimation on a space-time *** algorithm is applied to the numerical solution of the Navier-Stokes equation and is benchmarked to other implementations in the cases of the Taylor-Green vortex and Arnold-Beltrami-Childress flow.

关键词： fully nonlinear pdes systems of pdes Navier-Stokes equations Monte Carlo method deep neural network branching process random tree

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