THE HP-VERSION OF BEM-FAST CONVERGENCE,ADAPTIVITY AND EFFICIENT PRECONDITIONING
THE HP-VERSION OF BEM-FAST CONVERGENCE,ADAPTIVITY AND EFFICIENT PRECONDITIONING
摘要
In this survey paper we report on recent developments of the hp-version of the boundary element method （BEM）. As model problems we consider weakly singular and hypersingular integral equations of the first kind on a planar, open surface. We show that the Galerkin solutions （computed with the hp-version on geometric meshes） converge exponentially fast towards the exact solutions of the integral equations. An hp-adaptive algorithm is given and the implementation of the hp-version BEM is discussed together with the choice of efficient preconditioners for the ill-conditioned boundary element stiffness matrices. We also comment on the use of the hp-version BEM for solving Signorini contact problems in linear elasticity where the contact conditions are enforced only on the discrete set of Gauss-Lobatto points. Numerical results are presented which underline the theoretical results.
In this survey paper we report on recent developments of the hp-version of the boundary element method （BEM）. As model problems we consider weakly singular and hypersingular integral equations of the first kind on a planar, open surface. We show that the Galerkin solutions （computed with the hp-version on geometric meshes） converge exponentially fast towards the exact solutions of the integral equations. An hp-adaptive algorithm is given and the implementation of the hp-version BEM is discussed together with the choice of efficient preconditioners for the ill-conditioned boundary element stiffness matrices. We also comment on the use of the hp-version BEM for solving Signorini contact problems in linear elasticity where the contact conditions are enforced only on the discrete set of Gauss-Lobatto points. Numerical results are presented which underline the theoretical results.
参考文献20
-
1I. Babuska, B.-Q. Guo and E. P. Stephan, The h-p version of the boundary element method with geometric mesh on polygonal domains, Comput. Method. Appl. M., 80 (1990), 319-325.
-
2C. Carstensen, S. A. Funken and E. P. Stephan, A posteriori error estimates for hp-boundary element methods, Applicable Analysis, 61 (1996), 233-253.
-
3A. Chernov, S. Geyn, M. Maischak and E. P. Stephan, A boundary element method for contact problems with given friction for linear elasticity, Preprint IfAM, University of Hannover (2004).
-
4A. Bespalov and N. Heuer, The p-version of the boundary element method for weakly singular operators on piecewise plane open surfaces, Numer. Math., 106:1 (2007), 69-97.
-
5N. Heuer, F. Leydecker and E. P. Stephan, An iterative substructuring method for the hp-version of the BEM on quasi-uniform triangular meshes, Numer. Meth. Part. D. E., 23:4 (2007), 879-903.
-
6N. Heuer, M. Maischak and E. P. Stephan, Exponential convergence of the h-p-version for the boundary element method on open surfaces, Numer. Math., 83 (1999), 641-666.
-
7H. Holm, M. Maischak and E. P. Stephan, Exponential convergence of the h-p version bem for mixed boundary value problems on polyhedrons, Math. Method. Appl. Sci., 31, (2008), 2069-2093.
-
8N. Heuer, M. Mellado and E. P. Stephan, A p-adaptive algorithm for the BEM with the hypersingular operator on the plane screen, Int. J. Numer. Meth. Eng., 53:1 (2002), 85-104.
-
9M. Maischak and E. P. Stephan, The hp-version of the boundary element method in IR^3. The basic approximation results, Math. Method. Appl. Sci., 20 (1997), 461-476.
-
10M. Maischak and E. P. Stephan, Adaptive hp-versions of BEM for Signorini problems, Appl. Numer. Math., 54 (2005), 425-449.
-
1陈一鸣,郝亚娟,吴怡,刘若飞,张丽娜.Signorini接触问题的边界元方法及收敛性分析[J].燕山大学学报,2004,28(1):22-24. 被引量：2
-
2郝亚娟,陈一鸣,张丽娜,高宏伶,杨爱民.Signorini接触问题的边界变分不等式[J].数学理论与应用,2004,24(1):44-48. 被引量：2
-
3肖宏,郝亚娟,陈一鸣,张丽娜,刘若飞,吴怡.接触问题边界变分不等式解的存在唯一性[J].燕山大学学报,2004,28(1):1-6.
-
4陶利娜,谢资清.P-version间断有限元方法求解奇异摄动问题[J].云南师范大学学报（自然科学版）,2013,33(1):39-45.
-
5蔡碧野,向军.基于BEMD和PCA的数字图像压缩[J].计算机工程与应用,2011,47(23):185-187. 被引量：6
-
6史海旭.人生的双曲线[J].中国经济和信息化,2011(23):24-25.
-
7王卫东,冷岗松.Inequalities relating to L_p-version of Petty's conjectured projection inequality[J].Applied Mathematics and Mechanics(English Edition),2007,28(2):269-276. 被引量：1
-
8Houde Han Chunxiong Zheng.POISSON PRECONDITIONING FOR SELF-ADJOINT ELLIPTIC PROBLEMS[J].Journal of Computational Mathematics,2014,32(5):560-578.
-
9Wang, M,Zhang, S.THE OPTIMAL PRECONDITIONING IN THE DOMAIN DECOMPOSITION METHOD FOR WILSON ELEMENT[J].Journal of Computational Mathematics,1997,15(3):193-202.
-
10Jiwei Jia,Xiaozhe Hu,Jinchao Xu,Chen-Song Zhang.EFFECTS OF INTEGRATIONS AND ADAPTIVITY FOR THE EULERIAN-LAGRANGIAN METHOD[J].Journal of Computational Mathematics,2011,29(4):367-395. 被引量：1
