In the present paper we describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Neumann boundary conditions. It is well known that in such cases the solutions have singularities near the corners which poses a challenge for many existing methods. If the boundary data is smooth on each edge of the polygon, then in the vicinity of each corner the solution to the corresponding boundary integral equation has an expansion in terms of certain (analytically available) singular powers. Using the known behavior of the solution, universal discretizations have been constructed for the solution of the Dirichlet problem. However, the leading order behavior of solutions to the Neumann problem is $O(t^μ)$ for $μ\in (-1/2,0)$ depending on the angle at the corner (compared to $O(C+t^μ)$ with $μ>1/2$ for the Dirichlet problem); this presents a significant challenge in the design of universal discretizations. Our approach is based on using the discretization for the Dirichlet problem in order to compute a solution in the "weak sense" by solving an adjoint linear system; namely, it can be used to compute inner products with smooth functions accurately, but it cannot be interpolated. Furthermore we present a procedure to obtain accurate solutions arbitrarily close to the corner, by solving a sequence of small local subproblems in the vicinity of that corner. The results are illustrated with several numerical examples.
Displacement based finite element formulations over polygons: a comparison between Laplace interpolants, strain smoothing and scaled boundary polygon formulation
School of Civil & Environmental Engineering The University of New South Wales（新南威尔士大学土木与环境工程学院）
The Signorini problem for the Laplace operator is considered in a general polygonal domain. It is proved that the coincidence set consists of a finite number of boundary parts plus isolated points. The regularity of the solution is described. In particular, we show that the leading singularity is in general $r_i^{π/(2α_i)}$ at transition points of Signorini to Dirichlet or Neumann conditions but $r_i^{π/α_i}$ at kinks of the Signorini boundary, with $α_i$ being the internal angle of the domain at these critical points.
Analyticity and Stable Computation of Dirichlet-Neumann Operators for Laplace's Equation under Quasiperiodic Boundary Conditions in Two and Three Dimensions
Riesz potentials are well known objects of study in the theory of singular integrals that have been the subject of recent, increased interest from the numerical analysis community due to their connections with fractional Laplace problems and proposed use in certain domain decomposition methods. While the L$^p$-mapping properties of Riesz potentials on flat geometries are well-established, comparable results on rougher geometries for Sobolev spaces are very scarce. In this article, we study the continuity properties of the surface Riesz potential generated by the $1/\sqrt{x}$ singular kernel on…
Polar differentiation matrices for the Laplace equation in the disk subjected to nonhomogeneous Dirichlet, Neumann and Robin boundary conditions and the biharmonic equation subjected to nonhomogeneous Dirichlet conditions
In this paper we characterize the behavior of solutions to systems of boundary integral equations associated with Laplace transmission problems in composite media consisting of regions with polygonal boundaries. In particular we consider triple junctions, i.e. points at which three distinct media meet. We show that, under suitable conditions, solutions to the boundary integral equations in the vicinity of a triple junction are well-approximated by linear combinations of functions of the form $t^β,$ where $t$ is the distance of the point from the junction and the powers $β$ depend only on the m…
The orthogonality of Hilbert spaces whose elements can be represented as simple and double layer potentials is determined. Conditions of well-posed solvability of integral equations for the sum of simple and double layer potentials equivalent to double-sided Dirichlet, Neumann, and Dirichlet-Neumann boundary value problems for the Laplacian are established in the Hilbert space, elements of which as well as their normal derivatives have the jump through boundary surface. The properties of boundary operators that relate the double-sided boundary conditions of different types for the three-dimens…
We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter $\varepsilon>0$. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in $\varepsilon$. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due t…
High-Order Nyström/Convolution-Quadrature Solution of Time-Domain Scattering from Closed and Open Lipschitz Boundaries with Dirichlet and Neumann Boundary Conditions
A two-step method for solving planar Laplace problems via rational approximation is introduced. First complex rational approximations to the boundary data are determined by AAA approximation, either globally or locally near each corner or other singularity. The poles of these approximations outside the problem domain are then collected and used for a global least-squares fit to the solution. Typical problems are solved in a second of laptop time to 8-digit accuracy, all the way up to the corners, and the conjugate harmonic function is also provided. The AAA-least squares combination also offer…
The Helmholtz wave scattering problem by screens in 2D can be recast into first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introduce two new preconditioners, in the form of square-roots of local operators respectively for the corresponding problems with Dirichlet and Neumann conditions on the arc. They generalize the so-called "analytical" preconditioners available for Lipschitz scatterers. We introduce a functional setting adapted to the singularity of the problem and enabling the analysis of those preconditioners. The efficiency of the metho…
A parallel method for solving Laplace equations with Dirichlet data using local boundary integral equations and random walks
State Key Lab. of ASIC & System Fudan Univ.（复旦大学ASIC与系统国家重点实验室）；Dept. of Mathematics & Statistics University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校数学与统计系）
We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann opera…
The Green's functions for the Laplace equation respectively satisfying the Dirichlet and Neumann boundary conditions on the upper side of an infinite plane with a circular hole are introduced and constructed. These functions enables solution of the boundary value problems in domains where the hole is closed by any surface. This approach enables accounting for arbitrary positive and negative ground elevations inside the domain of interest, which, generally, is not possible to achieve using the regular method of images. Such problems appear in electrostatics, however, the methods developed apply…
