Meshless Virtual Boundary Method and Its Applications

The key of the numerical methods for solving differential equations and integral equations is to find a high accuracy approximated functions to replace the exact functions required. However, since approximated functions are usually based on the origin function, most approximation functions can not provide the same accuracy or high accuracy to the derivatives of the exact functions in various orders. The high accuracy numerical results of the lower order differential equations are usually disturbed by the errors of the derivatives of the exact functions. Sometimes, the errors between the approximation derivatives and exact derivatives are so heavily that numerical results are unavailable for higher order differential equations and integral equations while numerical methods are utilized. This paper proposed a novel quasi-interpolation scheme for numerical methods. The interpolation functions are based on an indeterminate integral procedure, thus the errors of the interpolated functions in various derivatives can be smoothed and high accuracy derivatives are available in desired order. Moreover, the interpolation scheme proposed is independent to the meshes usually used for building approximation functions and completing numerical integral in major numerical method, so the numerical method with the use of the interpolation scheme proposed has the characteristics of the meshless methods. Meanwhile, Green’s functions method is introduced and virtual boundaries which are corresponding to the completely close true boundaries of the domain to be solved. The system equations in integral form can be built according the conditions that the known boundaries values must be satisfied by responses of the virtual source functions existing on the virtual boundaries and the true source functions imposed on true boundaries. With orthogonal decomposition method, the discrete system equations based on the proposed approximated scheme can be solved easily even in large condition numbers. Three numerical examples included elastic problems and potential problems are calculated for illustrating the performances of the numerical method proposed. The results show that the MVBM is effectiveness and higher accuracy.