AN h-p VERSION OF THE CONTINUOUS PETROV-GALERKIN FINITE ELEMENT METHOD FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH SMOOTH AND NONSMOOTH KERNELS

作者：Yi, Lijun^*; Guo, Benqi

来源：SIAM Journal on Numerical Analysis, 2015, 53(6): 2677-2704.

DOI：10.1137/15M1006489

• Volterra integro-differential equations
• weakly singular kernel
• h-p version continuous Petrov-Galerkin method

We present an h-p version of the continuous Petrov-Galerkin (CPG) finite element method for linear Volterra integro-differential equations with smooth and nonsmooth kernels. We establish a priori error estimates in the L-2-, H-1-, and L-infinity-norms that are completely explicit with respect to the local discretization and regularity parameters. For singular solutions caused by the weakly singular kernels, we prove that algebraic convergence rates can be achieved for the h-p version of the CPG method with quasi-uniform meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometric time partitions and linearly increasing polynomial degrees. Numerical experiments are provided to illustrate the theoretical results.

• 出版日期2015
• 单位上海师范大学