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Mats G. Larson - One of the best experts on this subject based on the ideXlab platform.

Dominik Schotzau - One of the best experts on this subject based on the ideXlab platform.

  • Energy Norm a posteriori error estimation for hp adaptive discontinuous galerkin methods for elliptic problems in three dimensions
    Mathematical Models and Methods in Applied Sciences, 2011
    Co-Authors: Stefano Giani, Paul Houston, Dominik Schotzau
    Abstract:

    We develop the Energy Norm a posteriori error estimation for hp-version discontinuous Galerkin (DG) discretizations of elliptic boundary-value problems on 1-irregularly, isotropically refined affine hexahedral meshes in three dimensions. We derive a reliable and efficient indicator for the error measured in terms of the natural Energy Norm. The ratio of the efficiency and reliability constants is independent of the local mesh sizes and weakly depending on the polynomial degrees. In our analysis we make use of an hp-version averaging operator in three dimensions, which we explicitly construct and analyze. We use our error indicator in an hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples. Our numerical results indicate that exponential rates of convergence are achieved for problems with smooth solutions, as well as for problems with isotropic corner singularities.

  • Energy Norm a posteriori error estimation for divergence free discontinuous galerkin approximations of the navier stokes equations
    International Journal for Numerical Methods in Fluids, 2008
    Co-Authors: Guido Kanschat, Dominik Schotzau
    Abstract:

    We develop the Energy Norm a posteriori error analysis of exactly divergence-free discontinuous RTk/Qk Galerkin methods for the incompressible Navier–Stokes equations with small data. We derive upper and local lower bounds for the velocity–pressure error measured in terms of the natural Energy Norm of the discretization. Numerical examples illustrate the performance of the error estimator within an adaptive refinement strategy. Copyright © 2008 John Wiley & Sons, Ltd.

  • interior penalty discontinuous galerkin method for maxwell s equations Energy Norm error estimates
    Journal of Computational and Applied Mathematics, 2007
    Co-Authors: Marcus J. Grote, Anna Schneebeli, Dominik Schotzau
    Abstract:

    We develop the symmetric interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell equations in second-order form. We derive optimal a priori error estimates in the Energy Norm for smooth solutions. We also consider the case of low-regularity solutions that have singularities in space.

  • Energy Norm a posteriori error estimation of hp adaptive discontinuous galerkin methods for elliptic problems
    Mathematical Models and Methods in Applied Sciences, 2007
    Co-Authors: Paul Houston, Dominik Schotzau, Thomas P Wihler
    Abstract:

    In this paper, we develop the a posteriori error estimation of hp-version interior penalty discontinuous Galerkin discretizations of elliptic boundary-value problems. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) Energy Norm are derived. The bounds are explicit in the local mesh sizes and approximation orders. A series of numerical experiments illustrate the performance of the proposed estimators within an automatic hp-adaptive refinement procedure.

  • Discontinuous Galerkin Finite Element Method for the Wave Equation
    SIAM Journal on Numerical Analysis, 2006
    Co-Authors: Marcus J. Grote, Anna Schneebeli, Dominik Schotzau
    Abstract:

    The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second‐order wave equation. The resulting stiffness matrix is symmetric positive definite, and the mass matrix is essentially diagonal; hence, the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time‐ stepping scheme. Optimal a priori error bounds are derived in the Energy Norm and the L^2‐Norm for the semidiscrete formulation. In particular, the error in the Energy Norm is shown to converge with the optimal order {\cal O}(h^{\min\{s,\ell\}}) with respect to the mesh size h, the polynomial degree \ell, and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the L^2‐error is shown to converge with the optimal order {\cal O}(h^{\min\{s,\ell\}}). Numerical results confirm the expected convergence rates and illustrate the versatility of the method.

Martin Ehrendorfer - One of the best experts on this subject based on the ideXlab platform.

  • NOTES AND CORRESPONDENCE The Total Energy Norm in a Quasigeostrophic Model
    2020
    Co-Authors: Martin Ehrendorfer
    Abstract:

    Total Energy E as the sum of kinetic and available potential energies is considered here for quasigeostrophic (QG) dynamics. The discrete expression for E is derived for the QG model formulation of Marshall and Molteni. While E is conserved by the nonlinear unforced model equations, an analogous expression in terms of perturbed fields is, in general, not conserved for tangent-linearized versions of the model, thereby allowing for growth (or decay) in this total Energy Norm. Examples for structures linearly growing optimally (i.e., the so-called singular vectors) in terms of either the total Energy or just the kinetic Energy Norm are briefly illustrated and contrasted. It is argued that E might preferably be used (rather than kinetic Energy) in predictability and data assimilation studies that are based on the QG model considered here.

  • the total Energy Norm in a quasigeostrophic model
    Journal of the Atmospheric Sciences, 2000
    Co-Authors: Martin Ehrendorfer
    Abstract:

    Abstract Total Energy E as the sum of kinetic and available potential energies is considered here for quasigeostrophic (QG) dynamics. The discrete expression for E is derived for the QG model formulation of Marshall and Molteni. While E is conserved by the nonlinear unforced model equations, an analogous expression in terms of perturbed fields is, in general, not conserved for tangent-linearized versions of the model, thereby allowing for growth (or decay) in this total Energy Norm. Examples for structures linearly growing optimally (i.e., the so-called singular vectors) in terms of either the total Energy or just the kinetic Energy Norm are briefly illustrated and contrasted. It is argued that E might preferably be used (rather than kinetic Energy) in predictability and data assimilation studies that are based on the QG model considered here.

Peter Hansbo - One of the best experts on this subject based on the ideXlab platform.

F J Fuenmayor - One of the best experts on this subject based on the ideXlab platform.

  • a recovery explicit error estimator in Energy Norm for linear elasticity
    Computer Methods in Applied Mechanics and Engineering, 2015
    Co-Authors: E Nadal, J J Rodenas, Pedro Diez, F J Fuenmayor
    Abstract:

    Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the actual error. Typically, this has been achieved using residual-type estimates. One of the most popular and simpler (in terms of implementation) techniques used in commercial codes is the recovery-based error estimator. This technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in Energy Norm. Some attempts to remedy this situation provide bounds depending on unknown constants. Here, a new step towards obtaining error bounds from the recovery-based estimates is proposed. The idea is (1) to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error, (2) to add an explicit-type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and (3) to efficiently and accurately evaluate the constants appearing in the bounding expressions, thus providing asymptotic bounds. The numerical tests with h-adaptive refinement process show that the bounding property holds even for coarse meshes, providing upper bounds in practical applications. (C) 2015 Elsevier B.V. All rights reserved.

  • error estimation and error bounding in Energy Norm based on a displacement recovery technique
    ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering e-Book Full Papers, 2012
    Co-Authors: E Nadal, O A Gonzalezestrada, J J Rodenas, Stephane Bordas, F J Fuenmayor
    Abstract:

    Traditionally, recovery based error estimators have considered the evaluation of an enhanced stress field obtained from the raw Finite Element (FE) stress solution in linear elasticity. Instead of that, one can also obtain a recovered displacement field form the FE displacements [1]. Herein, we describe a superconvergent patch recovery of the displacement field which considers the local fulfilment of boundary and internal equilibrium equations, Dirichlet constraints and, for singular problems, the splitting of the displacement and stress fields into singular and smooth parts following the ideas presented in [2]. The development of this displacement recovery technique was motivated by the need to calculate the error in the displacement field required for the evaluation of upper bounds of the error in Energy Norm using a stress recovery technique [3, 4]. The accuracy of the proposed displacement recovery technique when used for error estimation in Energy Norm has been shown to be similar to that of the previously developed stress recovery technique. Although its computational cost is slightly higher than that of the stress recovery technique, the displacement recovery technique provides important advantages: • The recovered displacement solution can be projected to more refined meshes to obtain the initial displacements vector used by iterative solvers, thus reducing the number of iterations needed to reach the solution. • Practical upper error bounds can be directly obtained without using the projection techniques proposed in [3, 4]. • Adapting the ideas described in [5], we have also been able to evaluate accurate lower bounds of the error in Energy Norm. • The lower and practical upper bounds of the error in Energy Norm provided by the displacement recovery technique allowed for the development of error bounding techniques in quantities of interest as described in a paper presented in this session. Numerical tests based on the use of problems with known analytical solution have been used to validate the proposed techniques for error estimation and error bounding in Energy Norm.

  • accurate error estimate in Energy Norm using a nearly equilibrated kinematically admissible displacement recovery technique
    2012
    Co-Authors: E Nadal, O A Gonzalezestrada, J J Rodenas, Stephane Bordas, Pierre Kerfriden, F J Fuenmayor
    Abstract:

    In this paper we present a displacement recovery technique for linear elasticity problems solved with the Finite Element Method (FEM). The enhanced displacement eld provides a precise stress solution that can be used to obtain accurate estimates of the discretization error in Energy Norm using the Zienkiewicz and Zhu error estimator. Previous works [1,2] aimed at developing an upper bound of the error in Energy Norm were based on a stress recovery technique that induced small lacks of equilibrium which have to be accounted for using correction terms. These terms required approximations of the exact displacement error and were obtained using projection techniques which leaded to a higher computational eort. This