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We propose and study semidiscrete and fully discrete
finite element schemes based on appropriate relaxation models for
systems of Hyperbolic Conservation Laws.
These schemes are using piecewise polynomials of arbitrary degree and
their consistency error is of high order.
The methods are combined with an adaptive strategy that yields
fine mesh in shock regions and coarser mesh in the smooth parts of the
solution.
The computational performance of these methods is demonstrated by considering
scalar...
Long time simulations of transport equations raise computational challenges since they require both a large domain of calculation and sufficient accuracy. It is therefore advantageous, in terms of computational costs, to use a time varying adaptive mesh, with small cells in the region of interest and coarser cells where the solution is smooth. Biological models involving cell dynamics fall for instance within this framework and are often non conservative to account for cell division. In that case...
This work represents a first step towards the simulation of the
motion of water in a complex hydrodynamic configuration, such as
a channel network or a river delta, by means of a suitable
“combination” of different mathematical
models. In this framework a wide spectrum of space and time scales is involved
due to the presence of physical phenomena of
different nature.
Ideally, moving from a hierarchy of hydrodynamic models, one should solve
throughout the whole domain the most complex model (with...
These lecture notes present adaptive multiresolution schemes for evolutionary PDEs in Cartesian geometries. The discretization schemes are based either on finite volume or finite difference schemes. The concept of multiresolution analyses, including Harten’s approach for point and cell averages, is described in some detail. Then the sparse point representation method is discussed. Different strategies for adaptive time-stepping, like local scale dependent time stepping and time step control, are...
The concept of fully adaptive multiresolution finite volume schemes has been developed and investigated during the past decade. Here grid adaptation is realized by performing a multiscale decomposition of the discrete data at hand. By means of hard thresholding the resulting multiscale data are compressed. From the remaining data a locally refined grid is constructed. The aim of the present work is to give a self-contained overview on the construction of an appropriate multiresolution analysis using...
A finite volume method for the simulation of compressible aerodynamic flows is described.
Stabilisation and shock capturing is achieved by the use of an HLLC consistent numerical
flux function, with acoustic wave improvement. The method is implemented on an
unstructured hybrid mesh in three dimensions. A solution of higher order accuracy is
obtained by reconstruction, using an iteratively corrected least squares process, and by a
new limiting procedure....
In this paper, a dynamic viscoelastic problem is numerically studied. The variational problem is written in terms of the velocity field and it leads to a parabolic linear variational equation. A fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize time derivatives. An a priori error estimates result is recalled, from which the linear convergence is derived under suitable regularity conditions. Then, an a posteriori...
In this paper, a dynamic viscoelastic problem is numerically studied. The variational
problem is written in terms of the velocity field and it leads to a parabolic linear
variational equation. A fully discrete scheme is introduced by using the
finite element method to approximate the spatial variable and
an Euler scheme to discretize time derivatives. An a priori error estimates
result is recalled, from which the linear convergence is derived under suitable
regularity conditions. Then, an a posteriori
error...
The minimization of nonconvex functionals naturally arises in
materials sciences where deformation gradients in certain alloys exhibit
microstructures. For example, minimizing sequences of the nonconvex
Ericksen-James energy can be associated with deformations in
martensitic materials that
are observed in experiments[2,3].
— From the numerical
point of view, classical conforming and nonconforming finite element
discretizations have been observed to give minimizers
with their quality being highly
dependent...
We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell's system using limited boundary observations of the electric field in 3D. We derive a posteriori error estimates in the Tikhonov functional to be minimized and in the regularized solution of this functional, as well as formulate the corresponding adaptive algorithm. Our numerical experiments...
In this article we introduce an adaptive multi-level
method in space and time for convection diffusion problems. The scheme
is based on a multi-level spatial splitting and the use of different
time-steps. The temporal discretization relies on the characteristics method.
We derive an a posteriori error estimate and design a corresponding
adaptive algorithm.
The efficiency of the multi-level method is illustrated by numerical experiments,
in particular for a convection-dominated problem.
We discuss the occurrence of oscillations when using central schemes of the Lax-Friedrichs type (LFt), Rusanov’s method and the staggered and non-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema. The dependence of oscillatory properties on the numerical viscosity coefficient is investigated rigorously for the LFt schemes, illuminating also the properties of Rusanov’s method. It turns...
We discuss the occurrence of oscillations
when using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered and
non-staggered second order Nessyahu-Tadmor (NT) schemes.
Although these schemes are monotone or TVD, respectively,
oscillations may be introduced at local data extrema.
The dependence of oscillatory properties on the numerical viscosity
coefficient is investigated rigorously for the LFt schemes, illuminating also
the properties of Rusanov's method. It turns...
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