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We verify functional a posteriori error estimates proposed by S. Repin for a class of obstacle problems in two space dimensions. New benchmarks with known analytical solution are constructed based on one dimensional benchmark introduced by P. Harasim and J. Valdman. Numerical approximation of the solution of the obstacle problem is obtained by the finite element method using bilinear elements on a rectangular mesh. Error of the approximation is measured by a functional majorant. The majorant value...
Control strategies for nonlinear dynamical systems often make use of special system properties, which are, for example, differential flatness or exact input-output as well as input-to-state linearizability. However, approaches using these properties are unavoidably limited to specific classes of mathematical models. To generalize design procedures and to account for parameter uncertainties as well as modeling errors, an interval arithmetic approach for verified simulation of continuoustime dynamical...
This paper concerns accuracy-guaranteed numerical computations for linear systems. Due to the rapid progress of supercomputers, the treatable problem size is getting larger. The larger the problem size, the more rounding errors in floating-point arithmetic can accumulate in general, and the more inaccurate numerical solutions are obtained. Therefore, it is important to verify the accuracy of numerical solutions. Verified numerical computations are used to produce error bounds on numerical solutions....
This paper concerns the discretization of multiphase Darcy flows, in the case of
heterogeneous anisotropic porous media and general 3D meshes used in practice to represent
reservoir and basin geometries. An unconditionally coercive and symmetric vertex centred
approach is introduced in this paper. This scheme extends the Vertex Approximate Gradient
scheme (VAG), already introduced for single phase diffusive problems in [9], to multiphase
Darcy flows....
There are two mathematical models of elastic walls of healthy and atherosclerotic blood
vessels developed and studied. The models are included in a numerical model of global
blood circulation via recovery of the vessel wall state equation. The joint model allows
us to study the impact of arteries atherosclerotic disease of a set of arteries on
regional haemodynamics.
We consider mathematical models describing dynamics of an elastic beam which is clamped at its left end to a vibrating support and which can move freely at its right end between two rigid obstacles. We model the contact with Signorini's complementary conditions between the displacement and the shear stress. For this infinite dimensional contact problem, we propose a family of fully discretized approximations and their convergence is proved. Moreover some examples of implementation are presented....
The main objective of this paper is to prove
new necessary conditions to the existence of
KAM tori.
To do so, we develop a
set of
explicit a-priori estimates for smooth
solutions of Hamilton-Jacobi equations,
using a combination of methods from
viscosity solutions,
KAM and Aubry-Mather theories.
These estimates
are valid
in any
space dimension, and can be checked numerically
to detect gaps between KAM tori and Aubry-Mather sets.
We apply these results to detect non-integrable regions in
several...
We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG)
methods. The output of this detector is a reliably scaled, element-wise smoothness
estimate which is suited as a control input to a shock capture mechanism. Using an
artificial viscosity in the latter role, we obtain a DG scheme for the numerical solution
of nonlinear systems of conservation laws. Building on work by Persson and Peraire, we
thoroughly justify the...
Modern computer algebra software can be used to visualize vector fields. One of the most used is the Maple program. This program is used to visualize two and three-dimensional vector fields. The possibilities of plotting direction vectors, lines of force, equipotential curves and the method of colouring the surface area for two-dimensional cases are shown step by step. For three-dimensional arrays, these methods are applied to various slices of three-dimensional space, such as a plane or a cylindrical...
We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data...
We present a new methodology for the numerical resolution of the hydrodynamics
of incompressible viscid newtonian fluids. It is based on the Navier-Stokes
equations and we refer to it as the vorticity projection method.
The method is robust enough to handle complex and convoluted configurations
typical to the motion of biological structures in viscous fluids.
Although the method is applicable to three dimensions, we address here
in detail only the two dimensional case. We provide numerical data...
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