A sharp error estimate of piecewise polynomial collocation for nonlocal problems with weakly singular kernels

Abstract

As is well known, piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC) are used to approximate the weakly singular integrals $$\begin{equation*}I(a,b,x) =\int^b_a \frac{u(y)}{|x-y|^\gamma}\textrm{d}y, \quad x \in (a,b),\quad 0< \gamma <1,\end{equation*}$$which have local truncation errors $\mathcal{O} (h^2 )$ and $\mathcal{O} (h^{4-\gamma } )$, respectively. Moreover, for Fredholm weakly singular integral equations of the second kind, i.e., $\lambda u(x)- I(a,b,x) =f(x)$, $\lambda \neq 0$, the global convergence rates are also $\mathcal{O} (h^2 )$ and $\mathcal{O} (h^{4-\gamma } )$ by PLC and PQC in Atkinson (2009, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press). In this work we study the following nonlocal problems, which are similar to the above Fredholm integral equations: $$\begin{equation*}\int^b_a \frac{u(x)-u(y)}{|x-y|^\gamma}\textrm{d}y =f(x), \quad x \in (a,b),\quad 0< \gamma <1. \end{equation*}$$In the first part of this paper we prove that the weakly singular integrals $I(a,b,x)$ have optimal local truncation error $\mathcal{O}(h^4\eta _i^{-\gamma } )$ by PQC, where $\eta _i=\min \left \{x_i-a,b-x_i\right \}$ and $x_i$ coincides with an element junction point. Then the sharp global convergence orders $\mathcal{O}\left (h\right )$ and $\mathcal{O} (h^3)$ by PLC and PQC, respectively, are established for nonlocal problems. Finally, numerical experiments are shown to illustrate the effectiveness of the presented methods.

Fast algebraic multigrid for block-structured dense and Toeplitz-like-plus-Cross systems arising from nonlocal diffusion problems

“…Here the operator A denotes the block-structured systems (2.3), (2.9) and (2.10), respectively. It should be noted that the stability and convergence analysis of time-dependent nonlocal problem (2.2) and (2.4) can be seen in [1,20,19,11]. The convergence rate of the twogrid method for time-dependent block-structured systems (2.10) can be directly obtained by Theorem 4.10 and [22]: here we omit the related derivation.…”

“…Large, sparse, block-structured linear systems arise in a wide variety of applications throughout computational science and engineering involving advectiondiffusion flow [42], image process [32], Markov chains [43], Biot's consolidation model [36], Navier-Stocks equations and saddle point problems [7]. In this paper we study the fast algebraic multigrid for solving the block-structured dense linear systems, stemming from nonlocal problems [2,11,19,26] by the piecewise quadratic polynomial collocation approximations, whose associated matrix can be expressed as a 2 × 2 block structure (1.1)…”

“…In this work, the simple (traditional) restriction operator and prolongation operator are used in order to handle such block-structured dense systems (1.1), including nonsymmetric indefinite system, symmetric positive definite (SPD), Toeplitz-plus-diagonal systems, which derive from the nonlocal problems discussed in [2,11,19,26]. In general, it is still not at all easy for dense stiffness matrices [3,4,8,17], unless we can reduce the problem to the Toeplitz setting and we know the symbol, its zeros, and their orders [39].…”

“…Application in nonlocal model . Consider the time-dependent nonlocal diffusion problem, whose prototype is [2,6,11,19] (2.2)…”

Analysis of (shifted) piecewise quadratic polynomial collocation for nonlocal diffusion model

“…where 4 (Ω) with δ = rh and 0 < α < 1. Let u C (y) and u Q (y) be defined by (2.6) and (3.1), respectively.…”