TY - JOUR

T1 - Extended multidimensional integration formulas on polytope meshes

AU - Guessab, Allal

AU - Semisalov, Boris

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In this paper, we consider a general decomposition of any convex polytope P \subset \BbbR n into a set of subpolytopes \Omega i and develop methods for approximating a definite integral of a given function f over P when, rather than its values at some points, a number of integrals of f over the faces of \Omega i are only available. We present several new families of extended integration formulas that contain such integrals and provide in a special case of our result the multivariate analogues of midpoint, trapezoidal, Hammer, and Simpson rules. The paper also presents the best possible explicit constants for their approximation errors. Here we succeed in finding the connection between minimization of the global error estimate and construction of centroidal Voronoi tessellations of a given polytope with special density function depending on properties of the integrand. In the case of integrands with strong singularities, it leads to essential reduction of the error. These ideas were extended to a more general case, in which the domain is not necessary polytope and is not necessary convex. We perform numerical tests with integrands having steep gradients which allow the comparison of the new cubature formulas and show their accuracy and rates of convergence.

AB - In this paper, we consider a general decomposition of any convex polytope P \subset \BbbR n into a set of subpolytopes \Omega i and develop methods for approximating a definite integral of a given function f over P when, rather than its values at some points, a number of integrals of f over the faces of \Omega i are only available. We present several new families of extended integration formulas that contain such integrals and provide in a special case of our result the multivariate analogues of midpoint, trapezoidal, Hammer, and Simpson rules. The paper also presents the best possible explicit constants for their approximation errors. Here we succeed in finding the connection between minimization of the global error estimate and construction of centroidal Voronoi tessellations of a given polytope with special density function depending on properties of the integrand. In the case of integrands with strong singularities, it leads to essential reduction of the error. These ideas were extended to a more general case, in which the domain is not necessary polytope and is not necessary convex. We perform numerical tests with integrands having steep gradients which allow the comparison of the new cubature formulas and show their accuracy and rates of convergence.

KW - Approximation

KW - Best error estimates

KW - Centroidal Voronoi tessellation

KW - Convexity

KW - Cubature formulas

KW - Singularity of integrand

KW - approximation

KW - centroidal Voronoi tessellation

KW - convexity

KW - singularity of integrand

KW - best error estimates

KW - cubature formulas

KW - CONVEX

KW - NUMERICAL-INTEGRATION

UR - http://www.scopus.com/inward/record.url?scp=85074712261&partnerID=8YFLogxK

U2 - 10.1137/18M1234564

DO - 10.1137/18M1234564

M3 - Article

AN - SCOPUS:85074712261

VL - 41

SP - A3152-A3181

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 5

ER -