Weakly-singular Integral Equations Research Articles

Numerical algorithm with high accuracy for the modified Helmholtz equation with Robin boundary value problem

In order to obtain the numerical solutions of the modified Helmholtz equation with Robin boundary conditions in two dimensions, we have developed a numerical algorithm to solve the problem, which is based on a Nyström discretization. First, the problem is transformed to a weakly-singular integral equation. Then we construct a quadrature method that achieve a convergence rate of O(h3), which has the characteristics of simple calculation and high precision. Further, the convergence of the numerical solutions is proved based on the collectively compact operators theory and the single parameter asymptotic expansion of errors with odd power O(h3) is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. Finally, some numerical examples are presented to demonstrate the efficiency of the method.

Two Singularity Subtraction Schemes for a Class of Nonlinear Weakly Singular Integral Equations

Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and uses some finite dimensional linearization process to solve numerically the discrete problem. Its convergence is proved under mild hypotheses on the nonlinearity and the quadrature rule of the singularity subtraction scheme. The New Approach is based on linearization of the problem in its infinite dimensional setting, and discretization of the sequence of linear problems by singularity subtraction. It is more efficient than the former, as three numerical experiments confirm.

Adaptive BEM for elliptic PDE systems, part II: Isogeometric analysis with hierarchical B-splines for weakly-singular integral equations

We formulate and analyze an adaptive algorithm for isogeometric analysis with hierarchical B-splines for weakly-singular boundary integral equations. We prove that the employed weighted-residual error estimator is reliable and converges at optimal algebraic rate. Numerical experiments with isogeometric boundary elements for the 3D Poisson problem confirm the theoretical results, which also cover general elliptic systems like linear elasticity.

Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations

In the present work, we consider weakly-singular integral equations arising from linear second-order elliptic PDE systems with constant coefficients, including, e.g. linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract conditions for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee that the weighted-residual error estimator is reliable and converges at optimal algebraic rate if used within an adaptive algorithm. These conditions are satisfied, e.g. for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include the localization of (non-local) fractional Sobolev norms and local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.

Smoothing transformation for numerical solution of nonlinear weakly singular Volterra integral equations using quasilinearization and product integration methods

We consider extension of quasilinearization technique and method of lower and upper solutions for nonlinear weakly singular Volterra integral equations. We obtain linear weakly singular integral equations and discuss on their existence, uniqueness and regularity properties of a solution. On the other hand, we show that the solutions of these linear equations are quadratically convergent to the solution of nonlinear equation. Using smoothing transformation we regularize the linear equations and then to approximate their solutions we apply global product integration method. Because of employing quasilinearization technique and smoothing transformation we yield a sequence of small size linear algebraic systems in the discretization. Error analysis shows the error bound has two parts of quasilinearization error which is quadratically convergent and product integration error where by using smoothing transformation its convergence order is improved. The numerical results obtained from different numerical examples are in agreement with the theoretical results and comparison with the other methods confirms the efficiency of the proposed method.

Adaptive isogeometric boundary element methods with local smoothness control

In the frame of isogeometric analysis, we consider a Galerkin boundary element discretization of the hyper-singular integral equation associated with the 2D Laplacian. We propose and analyze an adaptive algorithm which locally refines the boundary partition and, moreover, steers the smoothness of the NURBS ansatz functions across elements. In particular and unlike prior work, the algorithm can increase and decrease the local smoothness properties and hence exploits the full potential of isogeometric analysis. We prove that the new adaptive strategy leads to linear convergence with optimal algebraic rates. Numerical experiments confirm the theoretical results. A short appendix comments on analogous results for the weakly-singular integral equation.

Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods

We define and analyze (local) multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the refinement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results.

Quadratic numerical treatment for singular integral equations with logarithmic kernel

The goal of this paper is to present a direct method for an approximative solution of a weakly singular integral equations (WSIE) with logarithmic kernel on a piecewise smooth integration path using a modified quadratic spline approximation, we also show that this approximation gives an efficient approach to the analytical solution of WSIE.

Adaptive BEM with inexact PCG solver yields almost optimal computational costs

We consider the preconditioned conjugate gradient method (PCG) with optimal preconditioner in the frame of the boundary element method for elliptic first-kind integral equations. Our adaptive algorithm steers the termination of PCG as well as the local mesh-refinement. Besides convergence with optimal algebraic rates, we also prove almost optimal computational complexity. In particular, we provide an additive Schwarz preconditioner which can be computed in linear complexity and which is optimal in the sense that the condition numbers of the preconditioned systems are uniformly bounded. As model problem serves the 2D or 3D Laplace operator and the associated weakly-singular integral equation with energy space widetilde{H}^{-1/2}(Gamma ). The main results also hold for the hyper-singular integral equation with energy space H^{1/2}(Gamma ).

A linear Uzawa-type FEM–BEM solver for nonlinear transmission problems

We propose a fully discrete Uzawa-type iteration for the Johnson–Nédélec formulation of a Laplace-type transmission problem with possible (strongly monotone) nonlinearity in the interior domain. In each step, we sequentially solve one BEM for the weakly-singular integral equation associated with the Laplace-operator and one FEM for the linear Yukawa equation. In particular, the nonlinearity is only evaluated to build the right-hand side of the Yukawa equation. The algorithm includes the inexact solution of the BEM/FEM part by a preconditioned CG method. We prove that the proposed method leads to linear convergence with respect to the number of Uzawa iterations. Moreover, while the current analysis of a direct FEM–BEM discretization of the Johnson–Nédélec formulation requires some restrictions on the ellipticity (resp. strong monotonicity constant) in the interior domain, our Uzawa-type solver avoids such assumptions.

Curvilinear mode-I/mode-II interface fracture with a curvature-dependent surface tension on the boundary

A new model of fracture mechanics considered previously by Sendova and Walton \cite{SendovaWalton2010}, Zemlyanova \cite{Zemlyanova2013}, and Zemlyanova and Walton \cite{Zemlyanova2012} is further developed on the example of a mixed mode curvilinear interface fracture located on the boundary of a partially debonded thin elastic inclusion embedded in an infinite thin elastic matrix. The effect of the nano-structure of the material is incorporated into the model in the form of a curvature-depended surface tension acting on the boundary of the fracture. It is shown that the introduction of the surface tension allows to eliminate the classical oscillating and power singularities of the order $1/2$ present in the linear elastic fracture mechanics. The mathematical methods used to solve the problem are based on the Muskhelishvili's complex potentials and the Savruk's integral representations. The mechanical problem is reduced to the system of singular integro-differential equations which is further reduced to a system of weakly-singular integral equations. The numerical computations and comparison with known results are presented.

Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations

In a recent work (Feischl et al. in Eng Anal Bound Elem 62:141–153, 2016), we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots. In the present work, we give a mathematical proof that this algorithm leads to convergence even with optimal algebraic rates. Technical contributions include a novel mesh-size function which also monitors the knot multiplicity as well as inverse estimates for NURBS in fractional-order Sobolev norms.

Adaptive boundary element methods for optimal convergence of point errors

One particular strength of the boundary element method is that it allows for a high-order pointwise approximation of the solution of the related partial differential equation via the representation formula. However, the high-order convergence and hence accuracy usually suffers from singularities of the Cauchy data. We propose two adaptive mesh-refining algorithms and prove their quasi-optimal convergence behavior with respect to an a posteriori computable bound for the point error in the representation formula. Numerical examples for the weakly-singular integral equations for the 2D and 3D Laplacian underline our theoretical findings.

Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations.

We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the unknown Galerkin BEM error. The required assumptions are weak and allow for piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. In particular, our analysis gives a first contribution to adaptive BEM in the frame of isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments underline the theoretical findings and show that the proposed adaptive strategy leads to optimal convergence.

Convergence of Adaptive BEM and Adaptive FEM-BEM Coupling for Estimators Without h-Weighting Factor

Abstract We analyze adaptive mesh-refining algorithms in the frame of boundary element methods (BEM) and the coupling of finite elements and boundary elements (FEM-BEM). Adaptivity is driven by the two-level error estimator proposed by Ernst P. Stephan, Norbert Heuer, and coworkers in the frame of BEM and FEM-BEM or by the residual error estimator introduced by Birgit Faermann for BEM for weakly-singular integral equations. We prove that in either case the usual adaptive algorithm drives the associated error estimator to zero. Emphasis is put on the fact that the error estimators considered are not even globally equivalent to weighted-residual error estimators for which recently convergence with quasi-optimal algebraic rates has been derived.

Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation

We analyze an adaptive boundary element method for Symm's integral equation in 2D and 3D which incorporates the approximation of the Dirichlet data $$g$$ g into the adaptive scheme. We prove quasi-optimal convergence rates for any $$H^{1/2}$$ H 1 / 2 -stable projection used for data approximation.

Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods

Abstract. We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive mesh-refinement is driven by the weighted-residual error estimator. By proving that this estimator is not only reliable, but under some regularity assumptions on the given data also efficient on locally refined meshes, we characterize the approximation class in terms of the Galerkin error only. In particular, this yields that no adaptive strategy can do better, and the weighted-residual error estimator is thus an optimal choice to steer the adaptive mesh-refinement. As a side result, we prove a weak form of the saturation assumption.

The modified Adomian decomposition method and the noise terms phenomenon for solving nonlinear weakly-singular Volterra and Fredholm integral equations

AbstractIn this paper, we use the systematic modified Adomian decomposition method (ADM) and the phenomenon of the self-canceling ”noise” terms for solving nonlinear weakly-singular Volterra, Fredholm, and Volterra-Fredholm integral equations. We show that the proposed approach minimizes the computation, when compared with other conventional schemes. Our results are validated by investigating several examples.

ON FULLY DISCRETE COLLOCATION METHODS FOR SOLVING WEAKLY SINGULAR INTEGRAL EQUATIONS

A popular class of methods for solving weakly singular integral equations is the class of piecewise polynomial collocation methods. In order to implement those methods one has to compute exactly certain integrals that determine the linear system to be solved. Unfortunately those integrals usually cannot be computed exactly and even when analytic formulas exist, their straightforward application may cause unacceptable roundoff errors resulting in apparent instability of those methods in the case of highly nonuniform grids. In this paper fully discrete analogs of the collocation methods, where integrals are replaced by quadrature formulas, are considered, corresponding error estimates are derived.

A Product Integration Approach Based on New Orthogonal Polynomials for Nonlinear Weakly Singular Integral Equations

This paper provides with a generalization of the work by Chelyshkov (Electron. Trans. Numer. Anal. 25(7): 17–26, 2006), who has introduced sequences of orthogonal polynomials over [0,1] which can be expressed in terms of Jacobi polynomials. We develop a new approach of product integration algorithm based on these orthogonal polynomials including the numerical quadratures for solving the nonlinear weakly singular Volterra integral equations. The convergence analysis of the proposed scheme is derived and numerical results are given showing a marked improvement in comparison with recent numerical methods.