A boundary element method for non-homogeneous biharmonic equations in non-smooth domains

Approximate solutions of non-homogeneous biharmonic equations with non-homogeneous boundary conditions in piecewise smooth domains are obtained. This work is based on reducing the original problem to an auxiliary boundary value problem (BVP) for the homogeneous biharmonic equation. After that, we employ the Nyström method to determine approximate solutions of the Sherman–Lauricella equation with a right-hand side derived from the boundary conditions of the auxiliary BVP. Note that the stability of the method depends on the invertibility of operators from a Toeplitz algebra of operators. These operators do not depend on the shape of the domain boundary but only on the opening angles of the boundary corners. Although there are no efficient criteria for their invertibility, it is known that the Nyström method for this equation has only a handful of such angles, so that it can be efficiently used in this case. Theoretical estimates show that for piecewise smooth boundaries, approximate solutions of the Sherman–Lauricella equation obtained by the Nyström method converge to the exact solution in 𝐿 2 -norm as 𝒪 ⁢ ( 1 / 𝑛 ) and the results of numerical experiments are consistent with the theoretical findings. Consequently, the approximate solutions of the Sherman–Lauricella equation are used to determine the solution of non-homogeneous biharmonic problems. Examples carried out for non-polygonal non-convex domains demonstrate the stability and a good convergence of the proposed method.