Continuous Solution for Boundary Value Problems on Non Rectangular Geometry

Boundary value problems on a rectangular domain can also be solved through non domain discretization techniques. These methods yield high order continuous solutions. One such method, based on Bézier functions and developed by the author, can solve linear, nonlinear, ordinary, partial, single, or coupled systems of differential equations, using the same consistent approach. In this paper the technique is extended to non-rectangular domains. This provides a mesh free alternate to the family of finite element or finite difference methods that are currently used to solve these problems. The problem requires no transformation and the setup is direct and simple. The solution is established by minimizing the error in the residuals of the differential equations and the error in the boundary conditions over the domain. In addition, the high order continuous solution is available in polynomial form. The solution is obtained through the application of standard unconstrained optimization. Three examples are discussed. The first example is Poisson’s equation in a circular region. The second is the solution to the Laplace equation over a five sided domain with one of the sides circular. The third example is a simple nonlinear extension of the differential equation in the second example. Directly accommodating a nonlinear boundary value problem, without any change, is the strength of this approach. Solutions are compared with ones from finite elements analysis where available. This approach can provide continuous solution to problems where previously only discontinuous solutions were available.