Integral Equations for Elliptic Problems with Edge Singularities and Applications to the Fourier-Boundary Element Method

摘要

Let u(x_1,x_2,z) = Σ_(l=1)~∞ u_l(x_1,x_2) sin (l_z) be the tangentially regular solution of a transmission problem for the Laplace operator in the horizontal strip R~2 * (0,π), the interface being lateral faces of a right prism of height π. We show that each Fourier coefficient u_l(x_1,x_2) is the single-layer potential V_l(x_1,x_2;q_l) relative to the two-dimensional Helmholtz operator -Δ + l~2. The density q_l solves a boundary integral equation, which is uniformly (with respect to the parameter l) well-posed in suitable Sobolev spaces. Decomposition of q_l into regular part and singular functions is investigated and used to design an optimally convergent discrete solution q_(l,h) of a boundary element method (BEM) the mesh size of which is explicitly graded and adapted. On the other hand, global regularity of q_l in suitable weighted Sobolev spaces is established. This is used to implement a more general optimally convergent BEM with solution q_(l,h) and with mesh size refined only near the corners of the interface. The truncated Fourier series u~(h,N)(x_1,x_2,z) = Σ_(l=1)~N V_e(x_1,x_2;q_(l,h)) sin(l_z), N ∈ N, is then a fully discrete solution of the transmission problem with optimal rate of convergence.