k -Version of Finite Element Method for BVPs and IVPs

The paper presents k -version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p -version hierarchical that permit use of any desired p -level without effecting global differentiability. HGDA/DG are true C i , C i j , C i j k , hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true C i , C i j k , C i j k approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C 0 . This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k -version of the finite element method is meaningless. Model problem studies in R 2 establish accuracy and superior convergence characteristics of true C i j p -version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p -convergence, k -convergence and p k -convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h , p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k -version and their limitations are discussed in the paper and are compared with present work.