749 research outputs found

    Computational Engineering

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    This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

    A POSTERIORI ANALYSIS OF AN ITERATIVE ALGORITHM FOR NAVIER-STOKES PROBLEM

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    This work deals with a posteriori error estimates for the Navier-Stokes equations. We propose a finite element discretization relying on the Galerkin method and we solve the discrete problem using an iterative method. Two sources of error appear, the discretization error and the linearization error. Balancing these two errors is very important to avoid performing an excessive number of iterations. Several numerical tests are provided to evaluate the efficiency of our indicators

    Element learning: a systematic approach of accelerating finite element-type methods via machine learning, with applications to radiative transfer

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    In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs). The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion. This map takes input of the element geometry and the PDEs' parameters on that element, and gives output of two operators -- (1) the in2out operator for inter-element communication, and (2) the in2sol operator (Green's function) for element-wise solution recovery. A significant advantage of this approach is that, once trained, this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining. Also, the training is significantly simpler since it is done on the element level instead on the entire domain. We call this approach element learning. This method is closely related to hybridizbale discontinuous Galerkin (HDG) methods in the sense that the local solvers of HDG are replaced by machine learning approaches. Numerical tests are presented for an example PDE, the radiative transfer equation, in a variety of scenarios with idealized or realistic cloud fields, with smooth or sharp gradient in the cloud boundary transition. Under a fixed accuracy level of 10−310^{-3} in the relative L2L^2 error, and polynomial degree p=6p=6 in each element, we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method

    Spectral Discontinuous Galerkin Method for Hyperbolic Problems

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    In this project we address the numerical approximation of hyperbolic equations and systems using the discontinuous Galerkin (DG) method in combination with higher order polynomial degrees. In short, this is called Spectral Discontinuous Galerkin (SDG) method. Our interest is to review the theoretical properties of the SDG-method, particularly for what concerns stability, convergence, dissipation and dispersion. Special emphases will be shed on the role of the two parameters,(the grid-size) and (the local polynomial degree). In this respect, we will carefully analyse the available theoretical results from the literature, then we extend some of them and implement several test cases with the purpose of assessing quantitatively the predicted theoretical properties

    Error analysis of a subgrid eddy viscosity multi-scale discretization of the Navier-Stokes equations

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    We propose a finite element discretization of the Navier–Stokes equations that relies on the variational multi-scale approach together with the addition of a Smagorinsky type viscosity, in order to take into account possible subgrid turbulence. We recall that the discrete problem admits a solution and prove a priori error estimates. Next we perform the a posteriori analysis of the discretization. Some numerical experiments justify the interest of this approach
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