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**Efficient adaptive integration of functions with sharp gradients and cusps in \(n\)-dimensional parallelepipeds.**
*(English)*
Zbl 1253.65034

Summary: In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a \(C^{0}\) function (derivative discontinuity at a point), a rate of convergence of \(n+1\) is obtained in \(\mathbb{R}^n\). Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function-a strongly localized function that is used for modeling sharp gradients. Then the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in \(n\)-dimensional parallelepiped elements.

### MSC:

65D30 | Numerical integration |

### Keywords:

adaptive quadrature; octree refinement; sharp gradients; cusps; partition-of-unity enrichment; density functional theory
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\textit{S. E. Mousavi} et al., Int. J. Numer. Methods Eng. 91, No. 4, 343--357 (2012; Zbl 1253.65034)

### References:

[1] | Patzák, Process zone resolution by extended finite elements, Engineering Fracture Mechanics 70 pp 957– (2003) |

[2] | Benvenuti, A regularized XFEM model for the transition from continuous to discontinuous displacements, International Journal for Numerical Methods in Engineering 74 pp 911– (2008) · Zbl 1158.74479 |

[3] | Benvenuti, A regularized XFEM framework for embedded cohesive interfaces, Computer Methods in Applied Mechanics and Engineering 197 pp 4367– (2008) · Zbl 1194.74364 |

[4] | Areias, Two-scale shear band evolution by local partition of unity, International Journal for Numerical Methods in Engineering 66 pp 878– (2006) · Zbl 1110.74841 |

[5] | Areias, Two-scale method for shear bands: thermal effects and variable bandwidth, International Journal for Numerical Methods in Engineering 72 pp 658– (2007) · Zbl 1194.74355 |

[6] | Tamma, Hierarchical p-version finite elements and adaptive a posteriori computational formulations for two-dimensional thermal analysis, Computers and Structures 32 (5) pp 1183– (1989) · Zbl 0705.73247 |

[7] | Merle, Solving thermal and phase change problems with the eXtended finite element method, Computational Mechanics 28 (5) pp 339– (2002) · Zbl 1073.76589 |

[8] | O’Hara, Generalized finite element analysis of three-dimensional heat transfer problems exhibiting sharp thermal gradients, Computer Methods in Applied Mechanics and Engineering 198 pp 1857– (2009) · Zbl 1227.80050 |

[9] | Brooks, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 32 (1-3) pp 199– (1982) · Zbl 0497.76041 |

[10] | Abbas, The XFEM for high-gradient solutions in convection-dominated problems, International Journal for Numerical Methods in Engineering 82 pp 1044– (2010) · Zbl 1188.76224 |

[11] | Kalashnikova, A discontinuous enrichment method for variable-coefficient advection-diffusion at high Péclet number, International Journal for Numerical Methods in Engineering 87 pp 309– (2011) · Zbl 1242.76125 |

[12] | Challacombe, Linear scaling computation of the Fock matrix. V. Hierarchical cubature for numerical integration of the exchange-correlation matrix, Journal of Chemical Physics 113 (22) pp 10037– (2000) |

[13] | Sukumar, Classical and enriched finite element formulations for Bloch-periodic boundary conditions, International Journal for Numerical Methods in Engineering 77 (8) pp 1121– (2009) · Zbl 1156.81313 |

[14] | Pask, Linear scaling solution of the all-electron Coulomb problem in solids, International Journal for Multiscale Computational Engineering, 2011, in press |

[15] | Melenk, The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099 |

[16] | Berntsen, Algorithm 720-an algorithm for adaptive cubature over a collection of 3-dimensional simplices, ACM Transactions on Mathematical Software 19 (3) pp 320– (1993) · Zbl 0890.65022 |

[17] | Gander, Adaptive quadrature-revisited, BIT Numerical Mathematics 40 (1) pp 84– (2000) · Zbl 0961.65018 |

[18] | Berntsen, Algorithm-706-DCUTRI: an algorithm for adaptive cubature over a collection of triangles, ACM Transactions on Mathematical Software 18 (3) pp 329– (1992) · Zbl 0892.65012 |

[19] | Genz, An adaptive numerical cubature algorithm for simplices, ACM Transactions on Mathematical Software 29 (3) pp 297– (2003) · Zbl 1072.65032 |

[20] | Lyness, Symmetric integration rules for hypercubes. III. Construction of integration rules using null rules, Mathematics of Computation 19 (92) pp 625– (1965) · Zbl 0142.12301 |

[21] | Berntsen, Error estimation in automatic quadrature routines, ACM Transactions on Mathematical Software 17 (2) pp 233– (1991) · Zbl 0900.65051 |

[22] | Tornberg, Multi-dimensional quadrature of singular and discontinuous functions, BIT Numerical Mathematics 42 (3) pp 644– (2002) · Zbl 1021.65010 |

[23] | Oh, The smooth piecewise polynomial particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particle methods, Computational Mechanics 40 pp 569– (2007) · Zbl 1165.74353 |

[24] | Ventura, On the elimination of quadrature subcells for discontinuous functions in the eXtended finite-element method, International Journal for Numerical Methods in Engineering 66 pp 761– (2006) · Zbl 1110.74858 |

[25] | Homeier, Numerical integration of functions with a sharp peak at or near one boundary using Möbius transformation, Journal of Computational Physics 87 pp 61– (1990) · Zbl 0693.65010 |

[26] | Homeier, On the evaluation of overlap integrals with exponential-type basis functions, International Journal of Quantum Chemistry 42 pp 761– (1992) |

[27] | López, Calculation of two-center exchange integrals with STOs using Möbius transformations, International Journal of Quantum Chemistry 49 pp 11– (1994) |

[28] | van Dooren, An adaptive algorithm for numerical integration over an n-dimensional cube, Journal of Computational and Applied Mathematics 2 (3) pp 207– (1976) · Zbl 0334.65024 |

[29] | Berntsen, An adaptive algorithm for the approximate calculation of multiple integrals, ACM Transactions on Mathematical Software 17 (4) pp 437– (1991) · Zbl 0900.65055 |

[30] | Pieper, Recursive Gauss integration, Communications in Numerical Methods in Engineering 15 (2) pp 77– (1999) · Zbl 0930.65013 |

[31] | Boyd, Chebyshev and Fourier spectral methods (2001) · Zbl 0994.65128 |

[32] | Legay, Strong and weak arbitrary discontinuities in spectral finite elements, International Journal for Numerical Methods in Engineering 64 pp 991– (2005) · Zbl 1167.74045 |

[33] | Cheng, Higher-order XFEM for curved strong and weak discontinuities, International Journal for Numerical Methods in Engineering 82 pp 564– (2010) · Zbl 1188.74052 |

[34] | Dréau, Studied X-FEM enrichment to handle material interfaces with higher order finite element, Computer Methods in Applied Mechanics and Engineering 199 pp 1922– (2010) · Zbl 1231.74406 |

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