Uploaded July 2026 | Updated September 2026, 2 weeks ago
Mixed finite element methods are capable of respecting physical conservation laws at the discrete level, by preserving the structure of the governing PDEs, however, this leads to computationally demanding saddle point systems with a large number of degrees of freedom. Robust and reliable solvers are therefore essential to make these methods feasible for large-scale problems, yet such solvers can introduce errors in conservation laws.
In this talk, Wietse M. Boon, University of Duisburg-Essen, considers a solver based on a decomposition of conventional finite element spaces, using spanning trees in the grid. This decomposition forms a new basis in which the mixed formulation of Darcy flow unravels from a large saddle-point problem into four smaller, symmetric positive definite systems. The resulting direct solver achieves a significant computational speed-up, without loss of accuracy. Dr. Boon proceeds by considering different variants of the solver. First, a model in which fractures are represented by subdomains of lower dimensionality, to show that the theory directly applies to this case and observe similar computational gains. Second, Dr. Boon will show the extension to a mixed formulation of linearized elasticity, in which the stress is an explicit variable in the system. Finally, this talk concludes with the implications for designing inexact solvers that respect physical constraints.
Watch *previous SIAM Geosciences webinars* at youtube.com/playlist?list=PLf_ipOSbWC86I1N9ATZJkPz1ym9KbPo5c
Learn more about *SIAM Activity Group on Geosciences* at siam.org/membership/activity-groups/detail/geosciences
#SIAM #Geosciences #MixedFinite #MixedDimentional #ElementMethods #Conservation #Decomposition #Elasticity
Mixed finite element methods are capable of respecting physical conservation laws at the discrete level, by preserving the structure of the governing PDEs, however, this leads to computationally demanding saddle point systems with a large number of degrees of freedom. Robust and reliable solvers are therefore essential to make these methods feasible for large-scale problems, yet such solvers can introduce errors in conservation laws.
In this talk, Wietse M. Boon, University of Duisburg-Essen, considers a solver based on a decomposition of conventional finite element spaces, using spanning trees in the grid. This decomposition forms a new basis in which the mixed formulation of Darcy flow unravels from a large saddle-point problem into four smaller, symmetric positive definite systems. The resulting direct solver achieves a significant computational speed-up, without loss of accuracy. Dr. Boon proceeds by considering different variants of the solver. First, a model in which fractures are represented by subdomains of lower dimensionality, to show that the theory directly applies to this case and observe similar computational gains. Second, Dr. Boon will show the extension to a mixed formulation of linearized elasticity, in which the stress is an explicit variable in the system. Finally, this talk concludes with the implications for designing inexact solvers that respect physical constraints.
Watch *previous SIAM Geosciences webinars* at youtube.com/playlist?list=PLf_ipOSbWC86I1N9ATZJkPz1ym9KbPo5c
Learn more about *SIAM Activity Group on Geosciences* at siam.org/membership/activity-groups/detail/geosciences
#SIAM #Geosciences #MixedFinite #MixedDimentional #ElementMethods #Conservation #Decomposition #Elasticity
