Scholarly Works - Book sections
Book section
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January 1, 2022
The goal of the first chapter is to present a brief motivation for the anisotropic mesh adaptation method and to illustrate its potential. We start the exposition by recalling several well-known facts concerning the numerical analysis of Galerkin (or finit ...
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January 1, 2022
We present an hp-variant of the anisotropic mesh adaptation methods discussed so far. That is, contrary to our discussion in Chap. 5, we now let the polynomial degree of approximation vary from mesh element to mesh element. This is essential in situations ...
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January 1, 2022
Employing the results from the previous chapter, we derive goal-oriented error estimates including the geometry of mesh elements. These estimates are based on interpolation error bounds. Furthermore, we present a goal-oriented anisotropic hp-mesh adaptatio ...
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January 1, 2022
We present some implementation aspects of the anisotropic hp-mesh adaptation algorithms. In particular, in Sect. 9.1, we present higher-order reconstruction techniques which are required by our algorithms for estimating of the interpolation error. Moreover ...
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January 1, 2022
We present several additional applications of the anisotropic hp-mesh adaptation methods to more practical problems. In particular, we deal with compressible flow acting on an isolated profile, time-dependent viscous shock-vortex interaction, and porous me ...
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January 1, 2022
We formulate the fundamental theoretical results which are later employed for the anisotropic mesh adaptation method. First, we recall the geometry terms of a mesh triangle K discussed in the previous chapter. Further, we define an interpolation of a suffi ...
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January 1, 2022
We discuss the construction of optimal meshes with respect to the interpolation error introduced in Chaps. 3 and 4. In particular, the goal is to construct a simplicial mesh such that the corresponding interpolation error is minimal, while the number of de ...
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January 1, 2022
In many practical applications, we are not interested in the solution u of the given partial differential equations as such, but in the value of a certain quantity of interest, which depends on the solution. ...
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January 1, 2022
We extend the theoretical results to the three-dimensional case. In the same spirit as for the two-dimensional case, we recall the geometry of a tetrahedron K and define the interpolation error function and the corresponding error estimates. The extension ...
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January 1, 2022
In this chapter, we formulate the initial problems related to the partition of the given domain Ω into a simplicial mesh. Furthermore, we recall the concept of the representation of simplicial meshes by a metric field, which is at the core of anisotropic m ...
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January 1, 2011
We review the current status of solution methods for nonlinear systems arising from high-order discretization of steady compressible flow problems. In this context, many of the difficulties that one faces are similar to, but more pronounced than, those tha ...
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