Numerical solution of an exterior Neumann problem using a double layer potential

Abstract

Abstract. We give here a variational formulation in H ' (T)/R of the exterior Neumann problem for the Laplace operator using a double layer potential. This formulation is then applied to the construction of a finite element method. Optimal error estimates are given.

Pointwise convergence of some boundary element methods. Part II

“…Operators of positive order also come up in boundary element Galerkin methods defined via hypersingular intégral operators ; see [7], [8], [9], [10], [12], [13], [14], [21], [22], [37], As an example and model operator let us consider the normal derivative of the double layer potential which is associated to Neumann problems, see [2], [7], [21], [38], ( Here, y n dénotes the fundamental solution of the Laplacian in 0? ", ft *\ n = 2 or n = 3, -and -are the normal derivatives with respect to the dn x dn y exterior unit normal to F at x and y, respectively, and K(x,y) is a sufficiently smooth kernel.…”

On the asymptotic convergence of collocation methods with spline functions of even degree

“…r Au = ß, f = z(r) on I\ have been used more recently in acoustics and electromagnetic fields and corresponding numerical treatments, [16], [17], [18], [35]. Employing the Cauchy-Riemann equations and integration by parts, (2.6) can be rewritten as whose principal part is given bỹ…”

“…For this equation preliminary convergence results can be found in [1], [2], [5], [14] and [33]. Our strongly elliptic systems with convolutional principal part contain, in addition, systems of integro-differential equations [3] (see [7]) with constant coefficients, certain singular integral equations, in particular, those of plane elasticity [7,Appendix], [24], [25], [26], [34], Fredholm integral equations of the second kind [6], [8], [11], [12], [13], [17], [27], [35], and also the integro-differential operator of Prandtl's wing theory [16], [17], [18], [24], [35].…”

Inverted finite elements: a new method for solving elliptic problems in unbounded domains

“…Reviewing the available strategies for solving PDEs in unbounded domains, we find a variety of methods having various degrees of accuracy, flexibility and sophistication. However, most of the existing methods rely -either on an integral representation of the exact solution and the use of Boundary Elements (see, e.g., [15,22,23,32,[34][35][36]); -or on replacing the unbounded domain by a sufficiently large bounded domain enclosed by a Perfectly Matched Layer (PML) (see [5,6]), or on the boundary of which an artificial boundary condition is prescribed; -or on a polar expansion of the solution like in spectral methods (see, e.g., [12,24]) or in infinite elements methods (see, e.g., [7,11,16,18,19,28]). …”