Understanding the influence of residual stresses on fracture behavior is a critical challenge. In this study, we propose a computational model for fracture analysis that incorporates the effects of residual stresses, utilizing the Particle Discretization Scheme Finite Element Method (PDS-FEM). In PDS-FEM, field variables are discretized using a pair of conjugate　geometries: Voronoi tessellations and Delaunay tessellations. Since the shape functions defined on these geometries are discontinuous and non-overlapping, the deformation of the solid continuum is represented as the translational motion of rigid body particles. Notably, this method provides the same level of accuracy in deformation analysis of solid continuum as conventional FEM with linear first-order elements, despite using discontinuous and non-overlapping shape functions. Thanks to these characteristics of PDS-FEM, we have successfully applied this method to fracture analysis of materials with residual stresses. In this presentation, we introduce the basic formulation of PDS-FEM and demonstrate its application to residual stress fields. We also present numerical analysis results for both quasi-static and dynamic crack propagation in residual stress fields, using desiccation cracks, thermal cracks, and tempered glass fractures as case studies. In each case, the numerical results are in excellent agreement with experimental observations.