Non-uniqueness of solution to quasi-1D compressible Euler equations
Three ways of interpolation on finite elements
Interpolation on finite elements usually occurs in a Hilbert space setting, which means that interpolation techniques involving orthogonal projection are an alternative for the traditional Lagrange nodal interpolation schemes. In addition to the Lagrange interpolation, this paper discusses the global orthogonal projection and the projection-based interpolation. These techniques are compared from the point of view of quality, efficiency, sensitivity to input parameters and other aspects. Local optimality...
On some a posteriori error estimation results for the method of lines
The paper is an attempt to present an (incomplete) historical survey of some basic results of residual type estimation procedures from the beginning of their development through contemporary results to future prospects. Recently we witness a rapidly increasing use of the -FEM which is due to the well-established theory. However, the conventional a posteriori error estimates (in the form of a single number per element) are not enough here, more complex estimates are needed, and this can be the way...
Discrete Green's function and maximum principles
In this paper the discrete Green’s function (DGF) is introduced and its fundamental properties are proven. Further it is indicated how to use these results to prove the discrete maximum principle for 1D Poisson equation discretized by the -FEM with pure Dirichlet or with mixed Dirichlet-Neumann boundary conditions and with piecewise constant coefficient.
Fieldless methods for the simulation of stationary and nonstationary induction heating
Space-time adaptive -FEM: Methodology overview
We present a new class of self-adaptive higher-order finite element methods (-FEM) which are free of analytical error estimates and thus work equally well for virtually all PDE problems ranging from simple linear elliptic equations to complex time-dependent nonlinear multiphysics coupled problems. The methods do not contain any tuning parameters and work reliably with both low- and high-order finite elements. The methodology was used to solve various types of problems including thermoelasticity,...
Description of the multi-dimensional finite volume solver EULER
This paper is aimed at the description of the multi-dimensional finite volume solver EULER, which has been developed for the numerical solution of the compressible Euler equations during several last years. The present overview of numerical schemes and the explanation of numerical techniques and tricks which have been used for EULER could be of certain interest not only for registered users but also for numerical mathematicians who have decided to implement a finite volume solver themselves. This...
Non-uniqueness of almost unidirectional inviscid compressible flow
Our aim is to find roots of the non-unique behavior of gases which can be observed in certain axisymmetric nozzle geometries under special flow regimes. For this purpose, we use several versions of the compressible Euler equations. We show that the main reason for the non-uniqueness is hidden in the energy decomposition into its internal and kinetic parts, and their complementary behavior. It turns out that, at least for inviscid compressible flows, a bifurcation can occur only at flow regimes with...
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