We develop numerical methods for analysis of fourth-order partial differential equations on domains with angular corners. For the finite element analysis of fourth-order partial differential equations, we have to use smoother basis functions whose derivatives are continuous. Since the derivatives of Lagrange basis functions for the conventional finite element method are not continuous, the complex Hermite basis functions are suggested. However, those existing exotic elements of Hermite type are complicated in construction and implementation. Whereas the approximation space for Isogeometric Analysis (IGA), developed recently, consists of B-spline basis functions with any desired regularity. However, IGA using single patch encounters difficulties in dealing with boundary value problems on irregular shaped polygonal domains. Inthis paper, in order to handle fourth-order problems with singularities, we introduce an Implicitly Enriched Galerkin method in which singular basis functions resembling the known point singularities are generated through a special geometric mapping and are combined with smooth basis functions through the at-top partition of unity (PU) functions. Unlike XFEM, this approach does not have singular integral problems. For the cases where multi-patches are necessary because of the complex geometry of the problems, it is difficult to join two patches along their interface in IGA. To end this, we combine the Implicitly Enriched Galerkin method with Schwarz domain decomposition methods. Thanks to Schwarz methods, we are able to break down the problems to smaller subproblems and are able to use different numerical techniques to solve each subproblem for localized treatment of complex geometries and singularities.Our aim in this research is to develop effective numerical methods with the less computational cost for the analysis of fourth-order problems on domains containing singularities. For this reason, we modify our method by applying different techniques such as Multicolor Schwarz and Supplemental Subdomain methods to reduce number of iterations for efficiency. Various numerical examples show the efficiency of our proposed method in dealing with fourth-order singular problems with crack singularities and/ or corner singularities.