A polynomial based iterative method for linear parabolic equations

Comparing Leja and Krylov Approximations of Large Scale Matrix Exponentials

“…[7,14]) as the methods of choice. Nevertheless, an alternative class of polynomial methods has been developed since the beginning (cf., e.g., [6,15,13]), which are based on direct interpolation or approximation of the exponential functions on the spectrum (or the field of values) of the relevant matrix. Despite of a preprocessing stage needed to get an estimate of some marginal eigenvalues, the latter are competitive with Krylov-like methods in several instances, namely on large scale, sparse and in general nonsymmetric matrices, arising from the spatial discretization of parabolic PDEs; see, e.g., [4,10,11].…”

The ReLPM Exponential Integrator for FE Discretizations of Advection-Diffusion Equations

“…The reference solution has been computed by CN with a local tolerance equal to 10 −6 , whereas the comparison of the errors is made using a local tolerance of 10 −4 for both methods (namely "tol" for the ReLPM algorithm in Table 1), which guarantees an error of the order of the spatial discretization error. Note that ReLPM is more accurate than CN at the final time, which shows that the mass-lumping technique does not significantly degrade the accuracy of the exponential integrator (5).…”

“…Such a system can be rewritten in the form ż = P −1 Hz + φ + P −1 q, t > 0 z(0) = c 0 which is suitable for the application of exponential integrators (cf. [5,6,7]). Observe that φ = P −1 b since we chose…”

“…[1,8]. Now we can consider the transformed system (4) with P −1 replaced by P −1 L , and apply the exact and explicit timemarching scheme (see the work by Schaefer [5] for FD spatial discretization)…”

“…Clearly, a key step in this procedure is given by estimating at low cost the reference focal interval for the spectrum of ∆tH L . Following [5] and [9], which deal with FD discretizations, we adopt the simplest estimate given directly by Gershgorin's theorem.…”