On the Numerical Solution of Parabolic Equations in a Single Space Variable by the Continuous Time Galerkin Method

Abstract

We consider the Galerkin method to solve a parabolic initial boundary value problem in one space variable, using piecewise polynomial functions and give an alternative proof of superconvergence. Then by means of Lobatto quadrature, we obtain purely explicit vector initial value problems without loss in the order of accuracy, global or pointwise.

A note onC o Galerkin methods for two-point boundary problems

“…The results of this paper can be easily applied to the case of two-point initial boundary problems (see [2]) and probably to other cases, such as nonlinear boundary problems. The findings of this paper stress the important part that Lobatto points play in the C ~ Galerkin method for two-point boundary problems.…”

The finite element method: A high‐performing approach for computing the probability of ruin and solving other ruin‐related problems

“…The errors on u AP (y, t) at time (t = T) is computed as follows: The above error measure takes into account the solution at all the points of the finite element mesh. However, as theoretically established by Bakker, 31 the FEM achieves superconvergence at those nodes that coincide with the boundaries of the FEM intervals (i.e., nodes 𝜂 1 , 𝜂 2 , … , 𝜂 Ne + 1 ). Then, to experimentally assess this, let us also consider the error measure:…”

“…In particular, using a standard personal computer, the solution of the ruin‐related problems considered can be obtained with an error (in the discrete maximum norm) of order 10 −5 or even much smaller in less than a second. Moreover, consistently with the theory developed in Bakker, 31 an extremely accurate (superconvergent) approximation is obtained at the boundaries of the finite elements, see Section 4. Thus, the FEM appears to be very suitable for solving ruin‐related differential problems and should be considered as a reliable and interesting alternative to other more traditional numerical approaches such as the finite difference method.…”

“…Specifically, three different kinds of polynomial basis are considered, namel, quadratic, cubic, and quartic polynomials, and following Bakker, 31 the integrals resulting from the FEM variational formulation are evaluated by Gauss–Lobatto integration, such to achieve superconvergence at the boundaries of the FEM elements. In addition, when dealing with the model by Costabile et al, 3 for the sake of computational efficiency a suitable operator splitting technique is employed, such that the FEM variational formulation can be used to discretize spatial derivatives only.…”

On superconvergence techniques

“…He used a semidiscrete Galerkin scheme (that is, the discretization in space only). With the help of the Laplace transform, an alternative proof of superconvergence is given in Bakker [16] and Adeboye [1]. See also Douglas [68] for a collocation method.…”