Higher order collocation scheme for differential equation

Abstract

Piecewise polynomial approximation of solution of differential equation with governing equation and its derivatives is proposed to enhance the computational efficiency by reducing total degree of freedom. Total degree of freedom can be reduced by assigning the governing equation and its derivatives to eliminate interior degrees of freedom, which are independent with boundary conditions. As a result, total degree of freedom with the proposed method is O(pn-1) if pth order polynomial shape functions are assigned to elements for n dimensional problem, while total degree of freedom with conventional p-refinement is O(pn) because dimension of boundary of domain is always smaller than that of the domain by one. Therefore, bottleneck region of FEM for large scale problem, Gauss-Jordan elimination of matrix with total degree of freedom, can be eased. On the other hand, shape functions of each element should be constructed independently, i.e. sacrifice efficiency for local stiffness matrixes for fast Gauss-Jordan elimination of global stiffness matrix. Basic concept and theories are described though one dimensional ordinary differential equation, error functions, as an example. Merits and demerits of formulations, weak form vs. strong form, are compared. Applications of the proposed method described for time integration scheme and plane stress problem. Detail derivation of the explicit time integration scheme is described and vibration of mass-spring system is presented as an example. In the case of plane stress problem, construction procedure of shape functions with the proposed method is described. The shape functions with HOC are assumed as nth order polynomial which satisfies compatibility and completeness. The shape functions show less mesh dependency because of its fast convergence rate. Moreover, adaptive refinement could be implemented in systematic way because compatibility can be satisfied although polynomial order of adjacent elements is different. Finally, locking problems, such as incompressible material or Mindlin plate element with small thickness could be handled with HOC. The proposed scheme does not suffer from spurious mode due to reduced integral because analytic integral does not induce locking problem.