### A comparison of multilevel adaptive methods for hurricane track prediction.

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Phase-field models, the simplest of which is Allen–Cahn’s problem, are characterized by a small parameter $\epsilon $ that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ${\epsilon}^{-2}$. Using an energy argument combined with a topological continuation argument and a spectral...

Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and...

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in ${\mathbb{R}}^{d}$, $d=2$ or 3, using backward Euler’s scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results...

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in ${\mathbb{R}}^{d}$, d=2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main...

We derive a posteriori estimates for a discretization in space of the standard Cahn–Hilliard equation with a double obstacle free energy. The derived estimates are robust and efficient, and in practice are combined with a heuristic time step adaptation. We present numerical experiments in two and three space dimensions and compare our method with an existing heuristic spatial mesh adaptation algorithm.

We address the numerical approximation of the two-phase Stefan problem and discuss an adaptive finite element method based on rigorous a posteriori error estimation and refinement/coarsening. We also investigate how to restrict coarsening for the resulting method to be stable and convergent. We review implementation issues associated with bisection and conclude with simulations of a persistent corner singularity, for which adaptivity is an essential tool.

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...

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....

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...

A new scheme for discretizing the P1 model on unstructured polygonal meshes is proposed. This scheme is designed such that its limit in the diffusion regime is the MPFA-O scheme which is proved to be a consistent variant of the Breil-Maire diffusion scheme. Numerical tests compare this scheme with a derived GLACE scheme for the P1 system.

We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly...

We discuss the issues of implementation of a higher order discontinuous Galerkin (DG) scheme for aerodynamics computations. In recent years a DG method has intensively been studied at Central Aerohydrodynamic Institute (TsAGI) where a computational code has been designed for numerical solution of the 3-D Euler and Navier-Stokes equations. Our discussion is mainly based on the results of the DG study conducted in TsAGI in collaboration with the NUMECA...

Structured adaptive mesh refinement (SAMR) techniques can enable cutting-edge simulations of problems governed by conservation laws. Focusing on the strictly hyperbolic case, these notes explain all algorithmic and mathematical details of a technically relevant implementation tailored for distributed memory computers. An overview of the background of commonly used finite volume discretizations for gas dynamics is included and typical benchmarks to quantify accuracy and performance of the dynamically...