The contribution considers parabolic PDEs describing uniphysics problems like nonstationary Darcy flow and their extension to multiphysics like poroelasticity problems. Discretization is assumed by mixed and standard/mixed finite elements in space and stable higher order methods in time. Parallelizable preconditioners for iterative solution of linear systems arising within the time steps are suggested and analysed. The analysis shows that the preconditioned systems are diagonalizable with very localized spectra. It indicates possible very fast convergence of Krylov type methods, which was also confirmed by numerical experiments with two step Radau time integration method.

1.
J. C.
Butcher
,
Integration Processes Based on Radau Quadrature Formulas
.
Mathematics of Computation
, Vol.
18
, No.
86
(Apr.,
1964
), pp.
233
244
.
2.
J. C.
Butcher
,
Implicit Runge-Kutta Processes Mathematics of Computation
Vol.
18
, No.
85
(Jan.,
1964
), pp.
50
64
.
3.
J. C.
Butcher
, Numerical Methods for Ordinary Differential Equations. Second Edition.
John Wiley
,
Chich-ester
2008
.
4.
O.
Axelsson
,
Global integration of differential equations through Lobatto quadrature
,
BIT
4
(
1964
), pp.
69
86
.
5.
O.
Axelsson
,
A class of A-stable methods
.
BIT
9
(
1969
), pp.
185
199
.
6.
O.
Axelsson
,
On the efficiency of a class of A-stable methods
.
BIT
14
(
1974
), pp.
279
287
.
7.
O.
Axelsson
,
R.
Blaheta
,
R.
Kohut
:
Preconditioned methods for high order strongly stable time integration methods with an application for a DAE problem
,
Numerical Linear Algebra with Applications
,
2015
(
22
), pp.
930
949
.
8.
O.
Axelsson
,
R.
Blaheta
and
T.
Luber
, Preconditioners for Mixed FEM Solution of Stationary and Nonstationary Porous Media Flow Problems.
Large-Scale Scientific Computing
(
Lirkov
,
I.
;
Margenov
,
S.
;
Waśniewski
,
J.
eds.),
Springer LNCS. Lecture Notes in Computer Science
.
9374
,
2016
, pp.
3
14
.
9.
R.
Blaheta
,
O.
Axelsson
,
T.
Luber
,
Preconditioning for systems arising in higher order Radau time discretization for PDE problems in multiphysics
.
In preparation
.
10.
O.
Axelsson
,
R.
Blaheta
,
Preconditioning of matrices partitioned in 2× 2 block form: Eigenvalue estimates and Schwarz DD for mixed FEM
.
Numer. Linear Algebra Appl.
17
(
2010
), pp.
787
810
.
11.
O.
Axelsson
,
R.
Blaheta
,
S.
Sysala
,
B.
Ahmad
,
On the solution of high order stable time integration methods
.
Boundary Value Problems
2013
(1), DOI: .
12.
D. E.
Keyes
 et al,
Multiphysics simulations: Challenges and opportunities
.
The International Journal of High Performance Computing Applications
27
(
2013
), pp.
4
83
.
13.
E.
Hairer
,
G.
Wanner
,
Stiff differential equations solved by Radau methods
.
Journal of Computational and Applied Mathematics
111
(
1999
)
93
111
.
14.
E.
Hairer
,
G.
Wanner
, Solving ordinary differential equations II: Stiff and differential-algebraic problems,
Berlin, New York
:
Springer-Verlag
,
1996
.
15.
Uri M.
Ascher
,
Linda R.
Petzold
, Computer methods for ordinary differential equations and differential-algebraic equations.
SIAM Philadelphia
,
1998
.
16.
K. E.
Brenan
,
S. L
Campbell
, L
R.
Petzold
, Numerical solution of initial-value problems in differential-algebraic equations.
Classics in applied mathematics, 14
.
SIAM Philadelphia
,
1996
.
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