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110
The Local Discontinuous Galerkin Method For TimeDependent ConvectionDiffusion Systems
"... In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, timedependent convectiondiffusion systems. These methods are an extension of the RungeKutta Discontinuous Galerkin methods for purely hyperbolic systems to convectiondiffusion systems and share with those methods the ..."
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Cited by 294 (31 self)
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In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, timedependent convectiondiffusion systems. These methods are an extension of the RungeKutta Discontinuous Galerkin methods for purely hyperbolic systems to convectiondiffusion systems and share with those methods their high parallelizability, their highorder formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L2stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.
The development of discontinuous Galerkin methods
, 1999
"... In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational ..."
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Cited by 176 (18 self)
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In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible NavierStokes equations, and HamiltonJacobilike equations.
A DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR HAMILTONJACOBI EQUATIONS
, 1998
"... In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear HamiltonJacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geo ..."
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Cited by 83 (13 self)
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In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear HamiltonJacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method.
Adaptive Local Refinement with Octree LoadBalancing for the Parallel Solution of ThreeDimensional Conservation Laws
 J. Parallel Distrib. Comput
, 1997
"... Conservation laws ae solved by a local Gaerkin finite element procedure with adapfive spacetime mesh refinement ad explicit time integration. The Courat stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to method ..."
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Cited by 67 (17 self)
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Conservation laws ae solved by a local Gaerkin finite element procedure with adapfive spacetime mesh refinement ad explicit time integration. The Courat stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to methods having a single global time step. Processor load imbalaces, introduced at adaptive enrichment steps, are corrected by using traversals of an octtee representing a spatial decomposition of the domain. To accommodate the variable time steps, octtee partitioning is extended to use weights derived from element size. Partition boundary smoothing reduces the communications volume of partitioning procedures for a modest cost. Computational results comparing parallel octtee ad inertial partitioning procedures ae presented for the threedimensional Euler equations of compressible flow solved on an IBM SP2 computer.
QuadratureFree Implementation Of The Discontinuous Galerkin Method For Hyperbolic Equations
 AIAA Journal
, 1996
"... Introduction Computational methods for aeroacoustics must possess accuracy properties that exceed those of conventional secondorder computational fluid dynamics (CFD) methods. At the same time, many problems of interest involve complex geometries that are not easily treated by common highorder met ..."
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Cited by 52 (15 self)
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Introduction Computational methods for aeroacoustics must possess accuracy properties that exceed those of conventional secondorder computational fluid dynamics (CFD) methods. At the same time, many problems of interest involve complex geometries that are not easily treated by common highorder methods that usually require a smooth, structured grid. In addition to the geometrically complex problem, we are particularly interested in strongly nonlinear flows that contain shock waves as a major source of sound generation, such as in the case of jet noise. In an effort to satisfy these requirements, the relatively untried discontinuous Galerkin (DG) method is being tested for hyperbolic problems. Some advantages of this approach include the ease with which the method can be applied to both structured and unstructured grids and its suitability for parallel computer architectures. The approach also has several useful mathematical properties. Johnson and Pitkarata 1 proved
An adaptive discontinuous galerkin technique with an orthogonal basis applied to compressible flow problems
 SIAM Review
"... Abstract. We present a highorder formulation for solving hyperbolic conservation laws using the Discontinuous Galerkin Method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit RungeKutta time discretization. Some results of higherorder adaptive refinement cal ..."
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Cited by 45 (4 self)
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Abstract. We present a highorder formulation for solving hyperbolic conservation laws using the Discontinuous Galerkin Method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit RungeKutta time discretization. Some results of higherorder adaptive refinement calculations are presented for inviscid Rayleigh Taylor flow instability and shock reflexion problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities. Key words. Discontinuous Galerkin, adaptive meshing, Orthogonal Basis.
Parallel Structures and Dynamic Load Balancing for Adaptive Finite Element Computation
 Applied Numerical Mathematics
, 1996
"... this paper, we have focused on describing and comparing several load balancing schemes. Comparisons by timing are difficult, since times vary between runs having the same parameters. The highspeed switch of the IBM SP2 computer is a shared resource that affects run times. More subtle effects can re ..."
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Cited by 40 (12 self)
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this paper, we have focused on describing and comparing several load balancing schemes. Comparisons by timing are difficult, since times vary between runs having the same parameters. The highspeed switch of the IBM SP2 computer is a shared resource that affects run times. More subtle effects can result from differences in the order in which messages used for migration are processed. Changes in the order in which those messages are received and integrated into the local MDB result in different traversal orders of the mesh entities. These differences cause small changes in load balancings and coarsenings. While such differences in meshes and partitionings do not affect the solution accuracy, they can cause sufficient changes in efficiency to make precise timings difficult. Qualitatively, PSIRB produced the best partitions (measured as a function of total analysis time). Octreegenerated partitions were comparable but resulted in slightly longer solution times. In both cases, one or two iterations of partition boundary smoothing led to a quality improvement. ITB by itself resulted in poorer partition quality, but is useful when mesh changes are small between computational stages. Predictive enrichment provided su21 perior performance to our current enrichment process with transient problems where there are frequent enrichment and balancing steps. Enhancements to the existing load balancing procedures and the implementation of new ones are under investigation. Improvements in the slicebyslice technique used by ITB for migration are necessary. Experiments with geometrical methods that use the spatial location of elements relative to the centroids of sending and receiving processors showed promise at reducing the number of processor interconnections. Vidwans et al. [39] pr...
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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Cited by 28 (0 self)
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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
A Hierarchical Partition Model for Adaptive Finite Element Computation
 Comput. Methods Appl. Mech. Engrg
, 1998
"... Introduction The finite element method (FEM) has become a standard analysis tool for solving partial differential equations (PDEs). Computationally demanding threedimensional problems make adaptive methods and parallel computation essential. Adaptive FEMs provide reliability, robustness, and time an ..."
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Cited by 27 (5 self)
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Introduction The finite element method (FEM) has become a standard analysis tool for solving partial differential equations (PDEs). Computationally demanding threedimensional problems make adaptive methods and parallel computation essential. Adaptive FEMs provide reliability, robustness, and time and space efficiency. In such a method, the computational domain is discretized into a mesh. During the adaptive solution process, portions of the mesh may be refined or coarsened (hrefinement) or moved to follow evolving phenomena (rrefinement). The method order may also be varied (prefinement). Each adaptive process concentrates the computational effort in areas where the solution resolution would otherwise be inadequate [7]. Conventional arraybased data representations, which work well for fixedmesh solutions, are not wellsuited to solutions involving mesh adaptivity [1]. Traversal of the data must be efficient in all cases, but w
Algorithms and data structures for massively parallel generic adaptive finite element codes
 ACM Trans. Math. Softw
, 2011
"... Today’s largest supercomputers have 100,000s of processor cores and offer the potential to solve partial differential equations discretized by billions of unknowns. However, the complexity of scaling to such large machines and problem sizes has so far prevented the emergence of generic software libr ..."
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Cited by 26 (12 self)
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Today’s largest supercomputers have 100,000s of processor cores and offer the potential to solve partial differential equations discretized by billions of unknowns. However, the complexity of scaling to such large machines and problem sizes has so far prevented the emergence of generic software libraries that support such computations, although these would lower the threshold of entry and enable many more applications to benefit from largescale computing. We are concerned with providing this functionality for meshadaptive finite element computations. We assume the existence of an “oracle ” that implements the generation and modification of an adaptive mesh distributed across many processors, and that responds to queries about its structure. Based on querying the oracle, we develop scalable algorithms and data structures for generic finite element methods. Specifically, we consider the parallel distribution of mesh data, global enumeration of degrees of freedom, constraints, and postprocessing. Our algorithms remove the bottlenecks that typically limit largescale adaptive finite element analyses. We demonstrate scalability of complete finite element workflows on up to 16,384 processors. An implementation of the proposed algorithms, based on the open source software p4est as mesh oracle, is provided