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Discontinuous Galerkin method for an evolution equation with a memory term of positive type. (English) Zbl 1198.65195

Summary: We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by \(C(t-s)^{\alpha-1}\), where \( 0<\alpha<1\). For the time discretization we apply the discontinuous Galerkin method using piecewise polynomials of degree at most \( q-1\), for \(q=1\) or \(2\). For the space discretization we use continuous piecewise-linear finite elements. The discrete solution satisfies an error bound of order \( k^q+h^2\ell(k)\), where \( k\) and \( h\) are the mesh sizes in time and space, respectively, and \( \ell(k)=\max(1,\log k^{-1})\). In the case \( q=2\), we prove a higher convergence rate of order \( k^3+h^2\ell(k)\) at the nodes of the time mesh. Typically, the partial derivatives of the exact solution are singular at \( t=0\), necessitating the use of non-uniform time steps. We compare our theoretical error bounds with the results of numerical computations.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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[1] Klas Adolfsson, Mikael Enelund, and Stig Larsson, Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 51-52, 5285 – 5304. · Zbl 1042.65103 · doi:10.1016/j.cma.2003.09.001
[2] Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu , Discontinuous Galerkin methods, Lecture Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, Berlin, 2000. Theory, computation and applications; Papers from the 1st International Symposium held in Newport, RI, May 24 – 26, 1999. · Zbl 0989.76045
[3] Eduardo Cuesta, Christian Lubich, and Cesar Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp. 75 (2006), no. 254, 673 – 696. · Zbl 1090.65147
[4] Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), no. 1, 43 – 77. · Zbl 0732.65093 · doi:10.1137/0728003
[5] Kenneth Eriksson, Claes Johnson, and Vidar Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 4, 611 – 643 (English, with French summary). · Zbl 0589.65070
[6] Stig Larsson, Vidar Thomée, and Lars B. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp. 67 (1998), no. 221, 45 – 71. · Zbl 0896.65090
[7] M. López-Fernández and C. Palencia, On the numerical inversion of the Laplace transform of certain holomorphic mappings, Appl. Numer. Math. 51 (2004), no. 2-3, 289 – 303. · Zbl 1059.65120 · doi:10.1016/j.apnum.2004.06.015
[8] María López-Fernández, César Palencia, and Achim Schädle, A spectral order method for inverting sectorial Laplace transforms, SIAM J. Numer. Anal. 44 (2006), no. 3, 1332 – 1350. · Zbl 1124.65120 · doi:10.1137/050629653
[9] J. C. López Marcos, A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal. 27 (1990), no. 1, 20 – 31. · Zbl 0693.65097 · doi:10.1137/0727002
[10] Ch. Lubich, I. H. Sloan, and V. Thomée, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comp. 65 (1996), no. 213, 1 – 17. · Zbl 0852.65138
[11] William McLean and Kassem Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math. 105 (2007), no. 3, 481 – 510. · Zbl 1111.65113 · doi:10.1007/s00211-006-0045-y
[12] W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive-type memory term, J. Austral. Math. Soc. Ser. B 35 (1993), no. 1, 23 – 70. · Zbl 0791.65105 · doi:10.1017/S0334270000007268
[13] William McLean and Vidar Thomée, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal. 24 (2004), no. 3, 439 – 463. · Zbl 1068.65146 · doi:10.1093/imanum/24.3.439
[14] W. McLean and V. Thomée, Numerical solution via Laplace transforms of a fractional order evolution equation, J. Integral Equations Appl., to appear. · Zbl 1195.65122
[15] W. McLean, V. Thomée, and L. B. Wahlbin, Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comput. Appl. Math. 69 (1996), no. 1, 49 – 69. · Zbl 0858.65143 · doi:10.1016/0377-0427(95)00025-9
[16] J. M. Sanz-Serna, A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal. 25 (1988), no. 2, 319 – 327. · Zbl 0643.65098 · doi:10.1137/0725022
[17] Achim Schädle, María López-Fernández, and Christian Lubich, Fast and oblivious convolution quadrature, SIAM J. Sci. Comput. 28 (2006), no. 2, 421 – 438. · Zbl 1111.65114 · doi:10.1137/050623139
[18] W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989), no. 1, 134 – 144. · Zbl 0692.45004 · doi:10.1063/1.528578
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