Locally Conservative B-spline Finite Element Methods for Two-point Boundary Value Problems

Abstract

The standard nodal Lagrangian based continuous Galerkin finite element method (FEM) and control volume finite element method (CVFEM) are well known techniques for solving partial differential equations. Both of these methods have a common shortcoming in that the first derivative of the approximate solution of both methods is discontinuous. Further shortcomings of nodal Lagrangian bases arise when considering time dependent problems. For instance, increasing the degree of the basis in an effort to improve the accuracy of the approximate solution prohibits the use of common techniques such as mass matrix lumping. We introduce a μth degree clamped basis-spline (B-spline) based analog of both the control volume finite element method and the continuous Galerkin finite element method in conjunction with a post processing technique which shall impose local conservation. The advantage of these techniques is that the B-spline basis is not only non-negative for any order μ, and thus lends itself to mass matrix lumping for higher order basis functions, but also, for μ > 2, each basis function is smooth on the domain. We implement both the B-spline based CVFEM and FEM techniques as well as the post processing technique as they pertain to solving various two-point boundary value problems. A comparison of the convergence rates and properties of the error associated with satisfying local conservation is presented.