Implicitly enriched Galerkin methods for numerical solutions of fourth‐order partial differential equations containing singularities

• Published in: Numerical methods for partial differential equations 2018-11, Vol.34 (6), p.2079-2112
• Main Authors: Kim, Sinae, Oh, Hae‐Soo, Palta, Birce, Kim, Hyunju
• Format: Article
• Language: English
• Subjects: Approximation; B spline functions; Basis functions; biharmonic equation; Boundary conditions; B‐spline (NURBS); Degrees of freedom; Domains; Elliptic differential equations; Elliptic functions; enriched isogeometric analysis(finite element method); Enrichment; Finite element method; Galerkin method; Galerkin method with B‐spline basis functions; Mapping; Methods; Numerical methods; Partial differential equations; Partition of Unity function with flat top; Partitions (mathematics); Singularity (mathematics); Unity
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.22277

Highlights are the following: For any integer n ≥ 0 , we construct C n ‐continuous partition of unity (PU) functions with flat‐top from B‐spline functions to have numerical solutions of fourth‐order equations with singularities. B‐spline functions are modified to satisfy clamped boundary conditions. To handle singularity arising in fourth‐order elliptic differential equations, these modified B‐spline functions are enriched either by introducing enrichment basis functions implicitly through particular geometric mappings or by adding singular basis functions explicitly. To show the effectiveness of the proposed implicit enrichment methods (mapping method), the accuracy, the number of degrees of freedom (DOF), and matrix condition numbers are computed and compared in the h‐refinement, the p‐refinement, and the k‐refinement of the approximation space of B‐spline basis functions. Using Partition of unity (PU) functions with flat‐top, B‐spline functions are modified to satisfy boundary conditions of the fourth‐order equations. Since the standard isogeometric analysis (IGA) as well as the conventional FEM have limitations in handling fourth‐order differential equations containing singularities, we consider two enrichment methods (explicit and implicit) in the framework of the p‐, the k, and the h‐refinements of IGA. We demonstrate that both enrichment methods yield good approximate solutions, but explicit enrichment methods give large (almost singular) matrix condition numbers and face integrating singular functions. Because of these limitations of external enrichment methods, we extensively investigate implicit enrichment methods (mapping methods) that virtually convert fourth‐order elliptic problems with singularities to problems with no influence of the singularities. Effectiveness of the proposed mapping method extensively tested to one‐dimensional fourth‐order equation with singularities. The implicit enrichment (mapping) method is extended to the two‐dimensional cases and test it to fourth‐order partial differential equations on cracked domains.