Analysis of finite element approximation and quadrature of Volterra integral equations

Abstract

In this article we study Galerkin finite element approximations to integral equations of the Volterra type. Our prime concern is the noncoercive case, which is not covered by the standard finite element theory. The question of rates of convergence is studied for the case where an exact stiffness matrix is available, as well as the case where the latter is approximated via quadrature rules. The optimality of these rules is also considered from the point of view of the effect the choice of the quadrature has on the overall rate of convergence.

A convergence result for a least-squares method using Schauder bases

“…Then inequality (1) gives that this sequence is also dense in (Y, · 2 ). Therefore, if Q n denotes the orthogonal projection of (Y, · 2 ) onto Dx 1 , .…”

Negative norm error control for second-kind convolution Volterra equations

“…However, this introduces a non-Galerkin error that is not bounded by either (6) or Theorem 1. This quadrature error is easy to take account of and we refer to Bedivan and Fix in [1] for problems of type (1), and also to Shaw and Whiteman in [6] for a similar treatment of an ODE with memory. Note that the controlled error e W −1 ∞ (J ) ≤ TOL is not the same as the error reported in the tables: e L∞(J ) ≤ S(T ) r L∞(J ) .…”

Numerical solution of linear Volterra integral equations of the second kind with sharp gradients

“…When applied to (16) and (18) this clusters points about the end of the integration range. We therefore obtain higher resolution in the vicinity of points where s ≈ t, as appear in the integrals (18).…”

“…When applied to (16) and (18) this clusters points about the end of the integration range. We therefore obtain higher resolution in the vicinity of points where s ≈ t, as appear in the integrals (18). One could continue to resolve the integrals (16) with uniform partitions, and only use graded partitions for the integrals (18), however, the results presented herein use the same graded quadrature partition spacing for all the integrals (16) and (18).…”

“…We therefore obtain higher resolution in the vicinity of points where s ≈ t, as appear in the integrals (18). One could continue to resolve the integrals (16) with uniform partitions, and only use graded partitions for the integrals (18), however, the results presented herein use the same graded quadrature partition spacing for all the integrals (16) and (18). We will subsequently refer to the HpPar method with element grading exponent, r, and partition grading exponent, r ′ , by the name HpPar r-r ′ .…”