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13
A Parallel hpAdaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws
 Appl. Numer. Math
, 1994
"... This paper describes a parallel algorithm based on discontinuous hpfinite element approximations of linear, scalar, hyperbolic conservation laws. The paper focuses on the development of an effective parallel adaptive strategy for such problems. Numerical experiments suggest that these techniques ..."
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This paper describes a parallel algorithm based on discontinuous hpfinite element approximations of linear, scalar, hyperbolic conservation laws. The paper focuses on the development of an effective parallel adaptive strategy for such problems. Numerical experiments suggest that these techniques are highly parallelizable and exponentially convergent, thereby yielding efficiency many times superior to conventional schemes for hyperbolic problems. 1
Adaptive Refinement Of Unstructured FiniteElement Meshes
"... The finite element method used in conjunction with adaptive mesh refinement algorithms can be an efficient tool in many scientific and engineering applications. In this paper we review algorithms for the adaptive refinement of unstructured simplicial meshes (triangulations and tetrahedralizations). ..."
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Cited by 19 (4 self)
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The finite element method used in conjunction with adaptive mesh refinement algorithms can be an efficient tool in many scientific and engineering applications. In this paper we review algorithms for the adaptive refinement of unstructured simplicial meshes (triangulations and tetrahedralizations). We discuss bounds on the quality of the meshes resulting from these refinement algorithms. Unrefinement and refinement along curved surfaces are also discussed. Finally, we give an overview of recent developments in parallel refinement algorithms. Key words. adaptive refinement, finiteelement meshes, parallel algorithms, unstructured meshes
Efficient Mesh Partitioning for Adaptive HP Finite Element Meshes
 In International Conference on Domain Decomposition Methods
, 1999
"... Introduction The use of domain decomposition solvers presumes the existence of a partitioning of the domain that distributes the computational effort equitably. Adaptive hp finite elements which are capable of delivering solution accuracies far superior to classical h\Gamma or p\Gammaversion finite ..."
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Introduction The use of domain decomposition solvers presumes the existence of a partitioning of the domain that distributes the computational effort equitably. Adaptive hp finite elements which are capable of delivering solution accuracies far superior to classical h\Gamma or p\Gammaversion finite element methods, for a given discretization size [BS94] create special difficulties in generating such load balanced partitions. Two major difficulties that arise in partitioning such meshes are a) the choice of a good a priori measure of computational effort, which can be equidistributed among the processors and b) minimizing the data migration among processors as the mesh changes and is repartitioned. In uniform meshes using simple solvers, computational effort is directly related to the degrees of freedom in each subdomain. In schemes using hp meshes, and domain decomposition solvers (e
A Survey of hpAdaptive Strategies for Elliptic Partial Differential Equations
"... The hp version of the finite element method (hpFEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations because it can achieve a convergence rate that is exponential in the number of degrees of freedom. hpFEM allows for refinement in ..."
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The hp version of the finite element method (hpFEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations because it can achieve a convergence rate that is exponential in the number of degrees of freedom. hpFEM allows for refinement in both the element size, h, and the polynomial degree, p. Like adaptive refinement for the h version of the finite element method, a posteriori error estimates can be used to determine where the mesh needs to be refined, but a single error estimate can not simultaneously determine whether it is better to do the refinement by h or by p. Several strategies for making this determination have been proposed over the years. In this paper we summarize these strategies and demonstrate the exponential convergence rates with two classic test problems.
Sparse Direct Factorizations through Unassembled HyperMatrices
 SUBMITTED TO COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
, 2007
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A posteriori error estimation of steadystate . . .
 COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
, 1998
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A Collection of 2D Elliptic Problems for Testing Adaptive Grid Refinement Algorithms ✩
"... Adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations (PDEs). The development of new algorithms and computer codes for the solution of PDEs usually involves the use of proofofconc ..."
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Adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations (PDEs). The development of new algorithms and computer codes for the solution of PDEs usually involves the use of proofofconcept test problems. 2D elliptic problems are often used as the first test bed for new algorithms and codes. This paper contains a set of twelve parametrized 2D elliptic test problems for adaptive grid refinement algorithms and codes. The problems exhibit a variety of types of singularities, near singularities, and other difficulties.
February 2010A Collection of 2D Elliptic Problems for Testing Adaptive Algorithms
"... Adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations (PDEs). The development of new algorithms and computer codes for the solution of PDEs usually involves the use of proofofconc ..."
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Adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations (PDEs). The development of new algorithms and computer codes for the solution of PDEs usually involves the use of proofofconcept test problems. 2D elliptic problems are often used as the first test bed for new algorithms and codes. This paper contains a set of twelve parametrized 2D elliptic test problems for adaptive grid refinement algorithms and codes. The problems exhibit a variety of types of singularities, near singularities, and other difficulties.
Patrick D. Gallagher, Under Secretary for Standards and Technology and DirectorA Comparison of hpAdaptive Strategies for Elliptic Partial Differential Equations (Long Version)
, 2011
"... The hp version of the finite element method (hpFEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations because it can achieve a convergence rate that is exponential in the number of degrees of freedom. hpFEM allows for refinement in ..."
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The hp version of the finite element method (hpFEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations because it can achieve a convergence rate that is exponential in the number of degrees of freedom. hpFEM allows for refinement in both the element size, h, and the polynomial degree, p. Like adaptive refinement for the h version of the finite element method, a posteriori error estimates can be used to determine where the mesh needs to be refined, but a single error estimate can not simultaneously determine whether it is better to do the refinement by h or by p. Several strategies for making this determination have been proposed over the years. In this paper we summarize these strategies and present the results of a numerical experiment to study the convergence properties of these strategies.