Error estimates for the gradient discretisation of degenerate parabolic equation of porous medium type

International audienceThe gradient discretisation method (GDM) is a generic framework for the spatial discretisation of partial differential equations. The goal of this contribution is to establish an error estimate for a class of degenerate parabolic problems, obtained under very mild regularity assumptions on the exact solution. Our study covers well-known models like the porous medium equation and the fast diffusion equations, as well as the strongly degenerate Stefan problem. Several schemes are then compared in a last section devoted to numerical results

A Priori Estimation of a Time Step for Numerically Solving Parabolic Problems

This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via explicit calculations. The selection strategy consists of the following two stages. At the first stage, we employ explicit calculations for selecting the appropriate time step. At the second stage, using the implicit scheme, we calculate the solution at a new time level. This solution should be close to the solution of our problem at this time level with a prescribed accuracy. Such an algorithm leads to explicit formulas for the calculation of the time step and takes into account both the dynamics of the problem solution and changes in coefficients of the equation and in its right-hand side. The same formulas for the evaluation of the time step are obtained by using a comparison of two approximate solutions, which are obtained using the explicit scheme with the primary time step and the step that is reduced by half. Numerical results are presented for a model parabolic boundary value problem, which demonstrate the robustness of the developed algorithm for the time step selection

Discontinuous Galerkin Method Applied to Navier-Stokes Equations

Discontinuous Galerkin (DG) finite element methods are becoming important techniques for the computational solution of many real-world problems describe by differential equations. They combine many attractive features of the finite element and the finite volume methods. These methods have been successfully applied to many important PDEs arising from a wide range of applications. DG methods are highly accurate numerical methods and have considerable advantages over the classical numerical methods available in the literature. DG methods can easily handle meshes with hanging nodes, elements of various types and shapes, and local spaces of different orders. Furthermore, DG methods provide accurate and efficient simulation of physical and engineering problems, especially in settings where the solutions exhibit poor regularity. For these reasons, they have attracted the attention of many researchers working in diverse areas, from computational fluid dynamics, solid mechanics and optimal control, to finance, biology and geology. In this talk, we give an overview of the main features of DG methods and their extensions. We first introduce the DG method for solving classical differential equations. Then, we extend the methods to other equations such as Navier-Stokes equations. The Navier-Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing

State Of the Art Report in the fields of numerical analysis and scientific computing. Final version as of 16/02/2020 deliverable D4.1 of the HORIZON 2020 project EURAD.: European Joint Programme on Radioactive Waste Management

Document information Project Acronym EURAD Project Title European Joint Programme on Radioactive Waste Management Project Type European Joint Programme (EJP) EC grant agreement No. 847593 Project starting / end date 1 st June 2019-30 May 2024 Work Package No. 4 Work Package Title Development and Improvement Of NUmerical methods and Tools for modelling coupled processes Work Package Acronym DONUT Deliverable No. 4.

1999 Flight Mechanics Symposium

This conference publication includes papers and abstracts presented at the Flight Mechanics Symposium held on May 18-20, 1999. Sponsored by the Guidance, Navigation and Control Center of Goddard Space Flight Center, this symposium featured technical papers on a wide range of issues related to orbit-attitude prediction, determination, and control; attitude sensor calibration; attitude determination error analysis; attitude dynamics; and orbit decay and maneuver strategy. Government, industry, and the academic community participated in the preparation and presentation of these papers

Non-local Allen-Cahn systems: Analysis and a primal dual active set method

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. Convergence of the primal-dual active set algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency