
Asymptotically compatible piecewise quadratic polynomial collocation for nonlocal model
In this paper, we propose and analyze piecewise quadratic polynomial col...
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Approximate solution of the integral equations involving kernel with additional singularity
The paper is devoted to the approximate solutions of the Fredholm integr...
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Adaptive BEM for elliptic PDE systems, part II: Isogeometric analysis with hierarchical Bsplines for weaklysingular integral equations
We formulate and analyze an adaptive algorithm for isogeometric analysis...
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Adaptive BEM for elliptic PDE systems, Part I: Abstract framework for weaklysingular integral equations
In the present work, we consider weaklysingular integral equations aris...
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Numerical solution to the 3D Static Maxwell equations in axisymmetric singular domains with arbitrary data
We propose a numerical method to solve the threedimensional static Maxw...
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Polynomial spline collocation method for solving weakly regular Volterra integral equations of the first kind
The polynomial spline collocation method is proposed for solution of Vol...
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Numerical methods for stochastic Volterra integral equations with weakly singular kernels
In this paper, we first establish the existence, uniqueness and Hölder c...
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A sharp error estimate of piecewise polynomial collocation for nonlocal problems with weakly singular kernels
As is well known, using piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC), respectively, to approximate the weakly singular integral I(a,b,x) =∫^b_a u(y)/xy^γdy, x ∈ (a,b) , 0< γ <1, have the local truncation error O(h^2) and O(h^4γ). Moreover, for Fredholm weakly singular integral equations of the second kind, i.e., λ u(x) I(a,b,x) =f(x) with λ≠ 0, also have global convergence rate O(h^2) and O(h^4γ) in [Atkinson and Han, Theoretical Numerical Analysis, Springer, 2009]. Formally, following nonlocal models can be viewed as Fredholm weakly singular integral equations ∫^b_a u(x)u(y)/xy^γdy =f(x), x ∈ (a,b) , 0< γ <1. However, there are still some significant differences for the models in these two fields. In the first part of this paper we prove that the weakly singular integral by PQC have an optimal local truncation error O(h^4η_i^γ), where η_i=min{x_ia,bx_i} and x_i coincides with an element junction point. Then a sharp global convergence estimate with O(h) and O(h^3) by PLC and PQC, respectively, are established for nonlocal problems. Finally, the numerical experiments including twodimensional case are given to illustrate the effectiveness of the presented method.
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