The least squares collocation method for the biharmonic equation in irregular and multiply-connected domains

Abstract

This paper reports new h-and p-versions of the least squares collocation method of high-order accuracy proposed and implemented for solving boundary value problems for the biharmonic equation in irregular and multiply-connected domains. This paper shows that approximate solutions obtained by the least squares collocation method converge with high order and agree with analytical solutions of test problems with high degree of accuracy. There has been a comparison made for the results achieved in this study and results of other authors who used finite difference and spectral methods.

On an Inverse Problem of Identifying an Unknown Boundary for the Biharmonic Equation from Cauchy Data

“…In mechanics, physics, and many engineering applications, the biharmonic equation is used as a governing equation to describe: the deformation of thin plates, the motion of fluids, free boundary problems, non-linear elasticity, and problems related to blending surfaces. Several implementations of the 2D biharmonic problem have been consecrated in simply and multiply-connected regions (see, e.g., [3,5,1,8,9,21,13,19,22,4,10,12]). Boundary conditions are essential and extremely important constraints for solving a boundary value problem.…”