Multivariate Birkhoff Interpolation

Lorentz, R. A.

doi:10.1007/bfb0088788

Cited by 72 publications

(56 citation statements)

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“…4(b)). With interpolation nodes chosen as above, the matrix is always invertible and the polynomial interpolation scheme is guaranteed to be stable ( [26,72]). …”

Section: Interpolation Of the Volume And Boundary Integralsmentioning

confidence: 99%

“…Finally, the jumps of the directional derivatives (n = 1, 2), are calculated in terms of the known jumps of partial derivatives, (70) and (71) For piecewise constant coefficient elliptic equations, recently Beale and Layton [68] proved that, with the correction formula computed as above for the standard five-point finite difference scheme, the solution to the corrected linear system (49) is second-order accurate in the discrete maximum norm, (72) For the general elliptic operator ℒ with possible anisotropy and inhomogeneity, a factor of log h will enter into the pointwise error estimate, which is stated in the following theorem.…”

Section: Derivation Of the Correction Formulamentioning

confidence: 99%

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A kernel-free boundary integral method for elliptic boundary value problems

Ying

Henriquez

2007

Journal of Computational Physics

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This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green's functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green's functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GM-RES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.

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“…4(b)). With interpolation nodes chosen as above, the matrix is always invertible and the polynomial interpolation scheme is guaranteed to be stable ( [26,72]). …”

Section: Interpolation Of the Volume And Boundary Integralsmentioning

confidence: 99%

Section: Derivation Of the Correction Formulamentioning

confidence: 99%

A kernel-free boundary integral method for elliptic boundary value problems

Ying

Henriquez

2007

Journal of Computational Physics

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“…, a n ) are the Gončarov polynomials. Such polynomials have an extensive history in numerical analysis, even for interpolation of derivatives [14]. Gončarov polynomials in two or more variables have the unusual property that the interpolation problem is solvable for any choice of the nodes of interpolation.…”

Section: Introductionmentioning

confidence: 99%

Multivariate delta Gončarov and Abel polynomials

Lorentz

Tringali

Yan

2017

Journal of Mathematical Analysis and Applications

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Classical Gončarov polynomials are polynomials which interpolate derivatives. Delta Gončarov polynomials are polynomials which interpolate delta operators, e.g., forward and backward difference operators. We extend fundamental aspects of the theory of classical bivariate Gončarov polynomials and univariate delta Gončarov polynomials to the multivariate setting using umbral calculus. After introducing systems of delta operators, we define multivariate delta Gončarov polynomials, show that the associated interpolation problem is always solvable, and derive a generating function (an Appell relation) for them. We show that systems of delta Gončarov polynomials on an interpolation grid Z ⊆ R d are of binomial type if and only if Z = AN d for some d × d matrix A. This motivates our definition of delta Abel polynomials to be exactly those delta Gončarov polynomials which are based on such a grid. Finally, compact formulas for delta Abel polynomials in all dimensions are given for separable systems of delta operators. This recovers a former result for classical bivariate Abel polynomials and extends previous partial results for classical trivariate Abel polynomials to all dimensions.

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“…However, such conditions are usually too restrictive and difficult to fulfill and apply. After all, the set of nodes for which Lagrange interpolation is not unique has measure zero, and interpolation is almost always possible; for literature and a historical account we refer to the recent survey [2] and the monograph [4], the latter one also containing a particularly extensive bibliography. Recently, a very interesting approach has been given by de Boor and Ron (see [1] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

On Multivariate Lagrange Interpolation

Sauer¹,

Xu²

1995

Mathematics of Computation

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Abstract. Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree « of a function /, which is a sum of integrals of certain (n + l)st directional derivatives of / multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.

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Multivariate Birkhoff Interpolation

Cited by 72 publications

References 0 publications

A kernel-free boundary integral method for elliptic boundary value problems

A kernel-free boundary integral method for elliptic boundary value problems

Multivariate delta Gončarov and Abel polynomials

On Multivariate Lagrange Interpolation

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