The P and H-P Versions of the Finite Element Method: Basic Principles and Properties

Babuška, Ivo; Suri, Manil

doi:10.21236/ada260237

Cited by 92 publications

(136 citation statements)

References 11 publications

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“…Then it satis es B DG (u v) = F DG (v) for all v 2 C b (M X). 4 We associate with each time interval I m an approximation order r m 0 and store these temporal orders in the vector r := fr m g M m=1 . The semidiscrete space in which we want to discretize (2.3), (2.4) in time is V r (M X) = fu : J !…”

Section: Dgfem For Abstract Parabolic Problemsmentioning

confidence: 99%

Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method

Schötzau¹,

Schwab²

2000

SIAM J. Numer. Anal.

181

163

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The Discontinuous Galerkin Finite Element Method (DGFEM) for the time discretization of parabolic problems is analyzed in a hp-version context. Error bounds which are explicit in the time step as well as the approximation order are derived and it is shown that the hp-DGFEM gives spectral convergence in problems with smooth time dependence. In conjunction with geometric time partitions it is proved that the hp-DGFEM results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms. For the h-version DGFEM algebraically graded time partitions are determined that give the optimal algebraic convergence rates. A fully discrete hp scheme is discussed exemplarily for the heat equation. The use of certain mesh-design principles for spatial discretizations yields exponential rates of convergence in time and space. Numerical examples con rm the theoretical results.

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Section: Dgfem For Abstract Parabolic Problemsmentioning

confidence: 99%

Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method

Schötzau¹,

Schwab²

2000

SIAM J. Numer. Anal.

181

163

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“…The higher-order ÿnite-element methods as compared to their lower-order counterpart, provide small di usion errors, easier implementation of the inf-sup condition and an exponential decay of the numerical error for smooth solutions (see Reference [1]). In practice, however, the existence of boundary layers restricts locally the smoothness of the solution and may a ect the overall accuracy.…”

Section: Introductionmentioning

confidence: 99%

On higher‐order mixed FEM for low Mach number flows: application to a natural convection benchmark problem

Heuveline

2003

Numerical Methods in Fluids

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SUMMARYWe consider higher-order mixed ÿnite elements with continuous pressures for the computation of stationary compressible ows at low Mach number. The proposed approach is based on a fully coupled treatment of the governing equations and therefore, for steady-state calculations, does not rely on timestepping techniques. The non-linear problem is solved by means of a quasi-Newton iteration. The strongly coupled system resulting from higher-order discretization of the linearized equations requires adequate solvers. We propose a new scheme based on multigrid methods with varying FEM ansatz orders on the grid hierarchy as well as multiplicative smoothers based on blocking techniques. Computational results are described for a benchmark conÿguration including a ow with heat transfer in the low Mach number regime. Furthermore, the issue of anisotropic grids is addressed in that context.

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“…Defining the notation for the exact errors, one obtains At this point, this equation is presented to simply observe that there is a relationship between the exact error E and the residual function d on the boundary. However, it is hoped that further analytical or numerical development of this approach can exploit equation (8).…”

Section: Boundary Error Estimate Approximate Solutionsmentioning

confidence: 99%

Hypersingular Residuals—a New Approach for Error Estimation in the Boundary Element Method

Paulino

Gray

Zarikian

1996

Int. J. Numer. Meth. Engng.

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SUMMARYThis paper presents a new approach for a posteriori 'pointwise' error estimation in the boundary element method. The estimator relies upon evaluation of the residual of hypersingular integral equations, and is therefore intrinsic to the boundary integral equation approach. A methodology is developed for approximating the error on the boundary as well as in the interior of the domain. Extensive computational experiments have been performed for the two-dimensional Laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve. Moreover, a reasonable estimate of the magnitude of the actual error is also predicted.

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The P and H-P Versions of the Finite Element Method: Basic Principles and Properties

Cited by 92 publications

References 11 publications

Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method

Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method

On higher‐order mixed FEM for low Mach number flows: application to a natural convection benchmark problem

Hypersingular Residuals—a New Approach for Error Estimation in the Boundary Element Method

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