Discretization of elliptic differential equations on curvilinear bounded domains with sparse grids

Dornseifer, Thomas; Pflaum, Christoph

doi:10.1007/bf02238512

Cited by 10 publications

(13 citation statements)

References 3 publications

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“…In the pioneering work of Zenger (1991) and Griebel (1991b), the foundations for adaptive refinement, multilevel solvers, and parallel algorithms for sparse grids were laid. Subsequent studies included the solution of the 3D Poisson equation Bungartz (1992aBungartz ( , 1992b, the generalization to arbitrary dimensionality d (Balder 1994) and to more general equations (the Helmholtz equation (Balder and Zenger 1996), parabolic problems using a time-space discretization (Balder, Rüde, Schneider and Zenger 1994), the biharmonic equation (Störtkuhl 1995), and general linear elliptic operators of second order in 2D (Pflaum 1996, Dornseifer andPflaum 1996). As a next step, the solution of general linear elliptic differential equations and, via mapping techniques, the treatment of more general geometries was implemented Dornseifer 1998, Dornseifer 1997) (see Figure 4.13).…”

Section: Sparse Grid Applications Pde Discretization Techniquesmentioning

confidence: 99%

Sparse grids

Bungartz¹,

Griebel²

2004

Acta Numerica

1,035

825

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We present a survey of the fundamentals and the applications of sparse grids, with a focus on the solution of partial differential equations (PDEs). The sparse grid approach, introduced in Zenger (1991), is based on a higherdimensional multiscale basis, which is derived from a one-dimensional multiscale basis by a tensor product construction. Discretizations on sparse grids involve O(N ·(log N ) d−1 ) degrees of freedom only, where d denotes the underlying problem's dimensionality and where N is the number of grid points in one coordinate direction at the boundary. The accuracy obtained with piecewise linear basis functions, for example, is O(N −2 · (log N ) d−1 ) with respect to the L 2 -and L ∞ -norm, if the solution has bounded second mixed derivatives. This way, the curse of dimensionality, i.e., the exponential dependence O(N d ) of conventional approaches, is overcome to some extent. For the energy norm, only O(N ) degrees of freedom are needed to give an accuracy of O(N −1 ). That is why sparse grids are especially well-suited for problems of very high dimensionality.The sparse grid approach can be extended to nonsmooth solutions by adaptive refinement methods. Furthermore, it can be generalized from piecewise linear to higher-order polynomials. Also, more sophisticated basis functions like interpolets, prewavelets, or wavelets can be used in a straightforward way.We describe the basic features of sparse grids and report the results of various numerical experiments for the solution of elliptic PDEs as well as for other selected problems such as numerical quadrature and data mining. CONTENTS1 Introduction 148 2 Breaking the curse of dimensionality 151 3 Piecewise linear interpolation on sparse grids 154 4 Generalizations, related concepts, applications 188 5 Numerical experiments 219 6 Concluding remarks 255 References 256 Breaking the curse of dimensionalityClassical approximation schemes exhibit the curse of dimensionality (Bellmann 1961) mentioned above. We havewhere r and d denote the isotropic smoothness of the function f and the problem's dimensionality, respectively. This is one of the main obstacles in the treatment of high-dimensional problems. Therefore, the question is whether we can find situations, i.e., either function spaces or error norms, for which the curse of dimensionality can be broken. At first glance, there is an easy way out: if we make a stronger assumption on the smoothness of the function f such thatOf course, such an assumption is completely unrealistic. However, about ten years ago, Barron (1993) found an interesting result. Denote by FL 1 the class of functions with Fourier transforms in L 1 . Then, consider the class of functions of R d with ∇f ∈ FL 1 .We expect an approximation rateMeanwhile, other function classes are known with such properties. These comprise certain radial basis schemes, stochastic sampling techniques, and approaches that work with spaces of functions with bounded mixed derivatives.A better understanding of these results is possible with the help of harmon...

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Section: Sparse Grid Applications Pde Discretization Techniquesmentioning

confidence: 99%

Sparse grids

Bungartz¹,

Griebel²

2004

Acta Numerica

1,035

825

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“…In this paper the discretization stencils were obtained by analytic calculations. If this is not possible, then one has to interpolate the variable coefficients by a piecewise constant interpolation of the variable coefficients on the sparse grid as in [13]. However, this paper was restricted to analytic calculation of the 27-stencils and 729-stencils.…”

Section: Numerical Resultsmentioning

confidence: 99%

A Sparse Grid Discretization with Variable Coefficient in High Dimensions

Hartmann,

Pflaum

2016

Preprint

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We present a Ritz-Galerkin discretization on sparse grids using pre-wavelets, which allows to solve elliptic differential equations with variable coefficients for dimension d = 2, 3 and higher dimensions d > 3. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality.This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple pre-wavelet stencil, and the classical operator dependent stencil for multilinear finite elements. Numerical simulation results are presented for a 3-dimensional problem on a curvilinear bounded domain and for a 6-dimensional problem with variable coefficients.Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below 10 using a standard diagonal preconditioner.

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“…Thus, we get a discrete solution u h of the equation (12). This approach for the discretization of elliptic equations on curvilinear bounded domains with sparse grids is explained in more detail in [3]. Figure 3 shows an adaptive sparse grid which was used for the discretization of equation (12).…”

Section: Numerical Example 1 Spectral Radius Of the Q-cycle In Case mentioning

confidence: 99%

“…The Q-cycle has been implemented for adaptive sparse grids and solves general second order elliptic differential equations. This algorithm was also applied for the discretization of elliptic differential equations on curvilinear bounded domains (see Dornseifer and Pflaum [3]). …”

Section: Introductionmentioning

confidence: 99%

A multilevel algorithm for the solution of second order elliptic differential equations on sparse grids

Pflaum

1998

Numerische Mathematik

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A multilevel algorithm is presented that solves general second order elliptic partial differential equations on adaptive sparse grids. The multilevel algorithm consists of several V-cycles in x -and y-direction. A suitable discretization provide that the discrete equation system can be solved in an efficient way. Numerical experiments show a convergence rate of order O(1) for the multilevel algorithm.

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Discretization of elliptic differential equations on curvilinear bounded domains with sparse grids

Cited by 10 publications

References 3 publications

Sparse grids

Sparse grids

A Sparse Grid Discretization with Variable Coefficient in High Dimensions

A multilevel algorithm for the solution of second order elliptic differential equations on sparse grids

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