Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations

Wang, Zixuan; Tang, Qi; Guo, Wei; Cheng, Yingda

doi:10.1016/j.jcp.2016.03.005

Cited by 43 publications

(83 citation statements)

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“…where n is the outward unit normal. The sparse grid IPDG scheme for (2.1) developed in [25] is defined as follows. We look for u h ∈ V k N , such that…”

Section: Ipdg Methods On Sparse Gridsmentioning

confidence: 99%

“…where σ is a positive penalty parameter, h = 2 −N is the uniform mesh size in each dimension. Note that the bilinear form B(·, ·) is the same as in [26,2], while instead of the expensive full grid piecewise polynomial space V k N , the more efficient sparse grid approximation space V k N is employed, leading to a significant reduction in the size of the linear algebraic system especially when d is large, see [25] for a detailed discussion.…”

Section: Ipdg Methods On Sparse Gridsmentioning

confidence: 99%

“…The details about the construction of the multiwavelet bases can be found in [1,25]. It is worth noting that space W k l can be constructed by means of projection operators as well, which will be helpful in the analysis below.…”

Section: Dg Finite Elementmentioning

confidence: 99%

“…Meanwhile, following the standard sparse grid philosophy, rather than including all the anisotropic tensor-product bases, only the ones with significant contribution to the approximation accuracy are chosen, leading to immense reduction in cost complexity when the dimension d is large. In particular, the number of DOF scales as O(h −1 | log h| d−1 ) instead of O(h −d ), where h denotes the mesh size in each direction; while the approximation property of the proposed sparse grid DG method is proven only slightly deteriorated for smooth solutions through both theoretical and numerical verifications, see [25,17].…”

mentioning

confidence: 98%

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The simulation of high-dimensional problems with manageable computational resource represents a long standing challenge. In a series of our recent work [25,17,18,24], a class of sparse grid DG methods has been formulated for solving various types of partial differential equations in high dimensions. By making use of the multiwavelet tensor-product bases on sparse grids in conjunction with the standard DG weak formulation, such a novel method is able to significantly reduce the computation and storage cost compared with full grid DG counterpart, while not compromising accuracy much for sufficiently smooth solutions.

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mentioning

confidence: 99%

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A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations

Guo¹,

Cheng²

2016

SIAM J. Sci. Comput.

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Abstract. In this paper, we develop a sparse grid discontinuous Galerkin (DG) scheme for transport equations and applied it to kinetic simulations. The method uses the weak formulations of traditional Runge-Kutta DG (RKDG) schemes for hyperbolic problems and is proven to be L 2 stable and convergent. A major advantage of the scheme lies in its low computational and storage cost due to the employed sparse finite element approximation space. This attractive feature is explored in simulating Vlasov and Boltzmann transport equations. Good performance in accuracy and conservation is verified by numerical tests in up to four dimensions.Key words. discontinuous Galerkin methods; sparse grid; high-dimensional transport equations; Vlasov equation; Boltzmann equation.1. Introduction. In this paper, we develop a sparse grid DG method for high-dimensional transport equations. High-dimensional transport problems are ubiquitous in science and engineering, and most evidently in kinetic simulations where it is necessary to track the evolution of probability density functions of particles. Deterministic kinetic simulations are very demanding due to the large computational and storage cost. To make the schemes more attractive comparing with the alternative probabilistic methods, an appealing approach is to explore the sparse grid techniques [6,15] with the aim of breaking the curse of dimensionality [4]. In the context of wavelets or sparse grid methods for kinetic transport equations, we mention the work of using wavelet-MRA methods for Vlasov equations [5], the combination technique for linear gyrokinetics [19], sparse adaptive finite element method [25], sparse discrete ordinates method [16] and sparse tensor spherical harmonics [17] for radiative transfer, among many others. This paper focuses on the DG method [13], which is a class of finite element methods using discontinuous approximation space for the numerical solution and the test functions. The RKDG scheme [14] developed in a series of papers for hyperbolic equations became very popular due to its provable convergence, excellent conservation properties and accommodation for adaptivity and parallel implementations. Recent years have seen great growth in the interest of applying DG methods to kinetic systems (see for example [3,18,20,9, 10]) because of the conservation properties and long time performance of the resulting simulations. However, the DG method is still deemed too costly in a realistic setting, often requiring more degrees of freedom than other high order numerical calculations.Recently, we developed a sparse grid DG method for high-dimensional elliptic problems [24]. A sparse DG finite element space has been constructed, reducing the degrees of freedom from the standardwhere h is the uniform mesh size in each dimension. The resulting scheme retains main properties of standard DG methods while making the computational cost tangebile for high-dimensional simulations. This motivates the current work for the transport equations, and we use kinetic problems as a...

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“…where n is the outward unit normal. The sparse grid IPDG scheme for (2.1) developed in [25] is defined as follows. We look for u h ∈ V k N , such that…”

Section: Ipdg Methods On Sparse Gridsmentioning

confidence: 99%