AN hp-VERSION DISCONTINUOUS GALERKIN METHOD FOR INTEGRO-DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

被引：45

作者：

Mustapha, K. ^{[1
]}

Brunner, H. ^{[2
,3
]}

Mustapha, H. ^{[4
]}

Schoetzau, D. ^{[5
]}

机构：

[1] King Fahd Univ Petr & Minerals, Dept Math & Stat, Dhahran 31261, Saudi Arabia

[2] Mem Univ Newfoundland, Dept Math & Stat, St John, NF A1C 5S7, Canada

[3] Hong Kong Baptist Univ, Dept Math, Kowloon Tong, Hong Kong, Peoples R China

[4] McGill Univ, Dept Min & Mat Engn, Montreal, PQ, Canada

[5] Univ British Columbia, Dept Math, Vancouver, BC V6T 1Z2, Canada

来源：

SIAM JOURNAL ON NUMERICAL ANALYSIS | 2011年 / 49卷 / 04期

基金：

加拿大自然科学与工程研究理事会;

关键词：

parabolic Volterra integro-differential equation; weakly singular kernel; hp-version DG time-stepping; exponential convergence; finite element method; fully discrete scheme; ORDINARY DIFFERENTIAL-EQUATIONS; ACCURATE NUMERICAL-METHOD; WEAKLY SINGULAR KERNEL; TIME DISCRETIZATION; EVOLUTION EQUATION; POSITIVE-TYPE; MEMORY TERM; APPROXIMATIONS;

D O I：

10.1137/100797114

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an hp-version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal hp-version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near t = 0 caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the h-version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems.

引用

页码：1369 / 1396

页数：28

共 28 条

[1]

[Anonymous], 2006, SPRINGER SER COMPUT

[2] hp-discontinuous Galerkin time-stepping for Volterra integrodifferential equations [J].