Author:
Ben-Artzi M.,Eidelman Y.,Fishelov D.
Abstract
AbstractThe paper incorporates new methods of numerical linear algebra for the approximation of the biharmonic equation with potential, namely, numerical solution of the Dirichlet problem for $$ \left( \frac{d}{dx}\right) ^4u(x)+c(x)u(x)=\phi (x),\quad 0<x<1. $$
d
dx
4
u
(
x
)
+
c
(
x
)
u
(
x
)
=
ϕ
(
x
)
,
0
<
x
<
1
.
High-order discrete finite difference operators are presented, constructed on the basis of discrete Hermitian derivatives, and the associated Discrete Biharmonic Operator (DBO). It is shown that the matrices associated with the discrete operator belong to a class of quasiseparable matrices of low rank matrices. The application of quasiseparable representation of rank structured matrices yields fast and stable algorithm for variable potentials c(x). Numerical examples corroborate the claim of high order accuracy of the algorithm, with optimal complexity O(N).
Funder
Hebrew University of Jerusalem
Publisher
Springer Science and Business Media LLC
Reference12 articles.
1. Ben-Artzi, M., Croisille, J.P., Fishelov, D.: Navier-Stokes Equations in Planar Domain. World Scientific, (2013)
2. Ben-Artzi, M., Croisille, J.P., Fishelov, D.: A fast direct solver for the biharmonic problem in a rectangular grid. SIAM J. Sci. Comput. 31, 303–333 (2008)
3. Ben-Artzi, M., Croisille, J.P., Fishelov, D., Katzir, R.: Discrete fourth-order Sturm-Liouville problems. IMA J. Numer. Anal. 38, 1475–1522 (2018)
4. Ben-Artzi, M., Katriel, G.: Spline functions, the biharmonic operator and approximate eigenvalues. Numer. Math. 141(4), 839–880 (2019)
5. Boito, P., Eidelman, Y.: Computations of quasiseparable representations of Green matrices. arXiv:2308 02701v1 [math.NA] 4 Aug 2023