Nyström Discretization of parabolic Boundary Integral Equations
نویسنده
چکیده
A Nyström method for the discretization of thermal layer potentials is proposed and analyzed. The method is based on considering the potentials as generalized Abel integral operators in time, where the kernel is a time dependent surface integral operator. The time discretization is the trapezoidal rule with a corrected weight at the endpoint to compensate for singularities of the integrand. The spatial discretization is a standard quadrature rule for surface integrals of smooth functions. We will discuss stability and convergence results of this discretization scheme for secondkind boundary integral equations of the heat equation. The method is explicit, does not require the computation of influence coefficients, and can be combined easily with recently developed fast heat solvers.
منابع مشابه
VARIATIONAL DISCRETIZATION AND MIXED METHODS FOR SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS WITH INTEGRAL CONSTRAINT
The aim of this work is to investigate the variational discretization and mixed finite element methods for optimal control problem governed by semi linear parabolic equations with integral constraint. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is not discreted. Optimal error estimates in L2 are established for the state...
متن کاملA Nyström-like Approach to Integral Equations with Singular Kernels
Traditional boundary element methods use panel-based discretization and exhibit low order convergence. In this paper, a new approach is proposed to discretize a singular integral equation. Global, numerically orthogonal bases are used to represent a solution, and mapping functions are used to represent the geometry. This method is capable of achieving spectral convergence, similar to the Nyströ...
متن کاملA Boundary Meshless Method for Neumann Problem
Boundary integral equations (BIE) are reformulations of boundary value problems for partial differential equations. There is a plethora of research on numerical methods for all types of these equations such as solving by discretization which includes numerical integration. In this paper, the Neumann problem is reformulated to a BIE, and then moving least squares as a meshless method is describe...
متن کاملHigh-order Accurate Nyström Discretization of Integral Equations with Weakly Singular Kernels on Smooth Curves in the Plane
Boundary integral equations and Nyström discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weaklysingular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarith...
متن کاملOn the Nyström Discretization of Integral Equations on Planar Curves with Corners
The Nyström method can produce ill-conditioned systems of linear equations when applied to integral equations on domains with corners. This defect can already be seen in the simple case of the integral equations arising from the Neumann problem for Laplace’s equation. We explain the origin of this instability and show that a straightforward modification to the Nyström scheme, which renders it m...
متن کامل