Developments in Isogeometric Analysis and Application to High Order Phase Field Models of Biomembranes Isogeometric analysis (IGA) is a numerical method for solving partial differential equations (PDEs) which was introduced with the aim of integrating finite element analysis with computer aided design systems. The main idea of the method is to use the same spline basis functions which describe the geometry in CAD systems for the approximation of solution fields in the finite element method (FEM). Originally NURBS which is a standard technology employed in CAD systems was adopted as basis functions in IGA but there were several variants of IGA using other technologies such as T splines PHT splines and subdivision surfaces as basis functions. In general IGA offers two key advantages over classical FEM: (i) by describing the CAD geometry exactly using smooth high order spline functions the mesh generation process is simplified and the interoperability between CAD and FEM is improved (ii) IGA can be viewed as a high order finite element method which offers basis functions with high inter element continuity and therefore can provide a primal variational formulation of high order PDEs in a straightforward fashion. The main goal of this thesis is to further advance isogeometric analysis by exploiting these major advantages namely precise geometric modeling and the use of smooth high order splines as basis functions and develop robust computational methods for problems with complex geometry and or complex multi physics. As the first contribution of this thesis we leverage the precise geometric modeling of isogeometric analysis and propose a new method for its coupling with meshfree discretizations. We exploit the strengths of both methods by using IGA to provide a smooth geometrically exact surface discretization of the problem domain boundary while the Reproducing Kernel Particle Method (RKPM) discretization is used to provide the volumetric discretization of the domain interior. The coupling strategy is based upon the higher order consistency or reproducing conditions that are directly imposed in the physical domain. The resulting coupled method enjoys several favorable features: (i) it preserves the geometric exactness of IGA (ii) it circumvents the need for global volumetric parameterization of the problem domain (iii) it achieves arbitrary order approximation accuracy while preserving higher order smoothness of the discretization. Several numerical examples are solved to show the optimal convergence properties of the coupled IGA–RKPM formulation and to demonstrate its effectiveness in constructing volumetric discretizations for complex geometry objects. As for the next contribution we exploit the use of smooth high order spline basis functions in IGA to solve high order surface PDEs governing the morphological evolution of vesicles. These governing equations are often consisted of geometric PDEs high order PDEs on stationary or evolving surfaces or a combination of them. We propose an isogeometric formulation for solving these PDEs. In the context of geometric PDEs we consider phase field approximations of mean curvature flow and Willmore flow problems and numerically study the convergence behavior of isogeometric analysis for these problems. As a model problem for high order PDEs on stationary surfaces we consider the Cahn–Hilliard equation on a sphere where the surface is modeled using a phase field approach. As for the high order PDEs on evolving surfaces a phase field model of a deforming multi component vesicle which consists of two fourth order nonlinear PDEs is solved using the isogeometric analysis in a primal variational framework. Through several numerical examples in 2D 3D and axisymmetric 3D settings we show the robustness of IGA for solving the considered phase field models. Finally we present a monolithic implicit formulation based on isogeometric analysis and generalized alpha time integration for simulating hydrodynamics of vesicles according to a phase field model. Compared to earlier works the number of equations of the phase field model which need to be solved is reduced by leveraging high continuity of NURBS functions and the algorithm is extended to 3D settings. We use residual based variational multi scale method (RBVMS) for solving Navier–Stokes equations while the rest of PDEs in the phase field model are treated using a standard Galerkin based IGA. We introduce the resistive immersed surface (RIS) method into the formulation which can be employed for an implicit description of complex geometries using a diffuse interface approach. The implementation highlights the robustness of the RBVMS method for Navier–Stokes equations of incompressible flows with non trivial localized forcing terms including bending and tension forces of the vesicle. The potential of the phase field model and isogeometric analysis for accurate simulation of a variety of fluid vesicle interaction problems in 2D and 3D is demonstrated.