We introduce an automatic variationally stable analysis (AVS) for finite element (FE) computations of scalar-valued convection-diffusion equations with non-constant and highly oscillatory coefficients. In the spirit o...
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ISBN:
(数字)9783030418007
ISBN:
(纸本)9783030417994
We introduce an automatic variationally stable analysis (AVS) for finite element (FE) computations of scalar-valued convection-diffusion equations with non-constant and highly oscillatory coefficients. In the spirit of least squares FE methods (Bochev and Gunzburger, Least-Squares Finite Element Methods, vol 166, Springer science & Business Media, Berlin, 2009), the AVS-FE method recasts the governing second order partial differential equation (PDE) into a system of first-order PDEs. However, in the subsequent derivation of the equivalent weak formulation, a Petrov-Galerkin technique is applied by using different regularities for the trial and test function spaces. We use standard FE approximation spaces for the trial spaces, which are C0, and broken Hilbert spaces for the test functions. Thus, we seek to compute pointwise continuous solutions for both the primal variable and its flux (as in least squares FE methods), while the test functions are piecewise discontinuous. To ensure the numerical stability of the subsequent FE discretizations, we apply the philosophy of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan (Comput Methods Appl Mech Eng 199(23):1558–1572, 2010;Discontinuous Petrov-Galerkin (DPG) method, Tech. rep., The Institute for computationalengineering and sciences, The University of Texas at Austin, 2015;SIAM J Numer Anal 49(5):1788–1809, 2011;Numer Methods Partial Differ Equ 27(1):70–105, 2011;Appl Numer Math 62(4):396–427,2012;Carstensen et al., SIAM J Numer Anal 52(3):1335–1353, 2014), by invoking test functions that lead to unconditionally stable numerical systems (if the kernel of the underlying differential operator is trivial). In the AVS-FE method, the discontinuous test functions are ascertained per the DPG approach from local, decoupled, and well-posed variational problems, which lead to best approximation properties in terms of the energy norm. We present various 2D numerical verifications, including convectio
Highlights of the asymptotic and numerical analysis of the steady axisymmetric swirl flow of a Newtonian liquid over a spinning disc and generated by a jet, impacting perpendicularly onto the latter in the direction o...
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We construct a new numerical method comprising upwind finite difference operators on asymptotically appropriate Shishkin meshes to obtain a numerical approximation to the solution of the Hemker problem. Numerical resu...
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The present study deals with the numerical simulation of a fluid–structure interaction problem. The fluid is represented by the incompressible Navier–Stokes equations and the structure is described by an ODE dependi...
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In this note we examine the a priori and a posteriori analysis of discontinuous Galerkin finite element discretisations of semilinear elliptic PDEs with polynomial nonlinearity. We show that optimal a priori error bou...
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The Distributed and Unified Numerics Environment (Dune) is a set of open-source C++ libraries for the implementation of finite element and finite volume methods. Over the last 15 years it has become one of the most co...
ISBN:
(数字)9783030597023
ISBN:
(纸本)9783030597016
The Distributed and Unified Numerics Environment (Dune) is a set of open-source C++ libraries for the implementation of finite element and finite volume methods. Over the last 15 years it has become one of the most commonly used libraries for the implementation of new, efficient simulation methods in science and engineering. Describing the main Dune libraries in detail, this book covers access to core features like grids, shape functions, and linear algebra, but also higher-level topics like function space bases and assemblers. It includes extensive information on programmer interfaces, together with a wealth of completed examples that illustrate how these interfaces are used in practice. After having read the book, readers will be prepared to write their own advanced finite element simulators, tapping the power of Dune to do so.
We perform numerical experiments on one-dimensional singularly perturbed problems of reaction-convection-diffusion type, using isogeometric analysis. In particular, we use a Galerkin formulation with B-splines as basi...
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In this paper we consider a singularly perturbed parabolic problem of reaction-diffusion type, posed on a two dimensional domain in space. We focus our attention on the nature of the singularity in the solution, cause...
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