Lecture series on MHM methods
Frédéric Valentin (LNCC, Petrópolis, Brazil) and
Antonio Tadeu Azevedo Gomes (LNCC, Petrópolis, Brazil) will present in a series of five lectures
the principles and practical implementation aspects of the MHM method.
These lectures will take place at the Laboratoire Dieudonné, Campus Valrose, Nice.
Abstract
Multiscale finite element methods approach the solution of partial differential equations with heterogeneous coefficients on coarse partitions,
and are becoming an attractive alternative to classical finite element methods that require very fine meshes.
This five-lecture course will present an overview of the origin of the Multiscale Hybrid-Mixed method (MHM for short) as well as
the MHM's basic theory and practical aspects of the implementation of the MHM method. The MHM is a byproduct of a general methodology
that uses the hybridization procedure on the continuous level to characterize the unknowns as a direct sum of a ``coarse'' global solution
and the solutions of local Neumann problems. At the discrete level, the local problems drive the multiscale basis functions,
while the global ``coarse'' problem is responsible for the computation of the (face) degrees of freedom. Since the local problems are entirely independent,
the method takes full advantage of parallel computations. The MHM method has exciting features such as super-convergence,
robustness in terms of (small) physical parameters, and local conservation properties on general polytopal meshes. Also, it provides a face-based a posteriori error
estimator which drives adaptative strategies. Those combined features make the method an affordable and highly precise strategy to solve massive multiscale problems
in complicated domains. The MHM method also shares similarities (and differences) with other multiscale and domain decomposition methods. All those aspects will be
illustrated during this course through numerical simulations using the MultiScale FEM Library(MSL) from the IPES Research Group (http://ipes.lncc.br/)
at the National Laboratory for Scientific Computing (LNCC), Brazil. The participants will at the end of these lectures have a broad and
precise overview of the fundamental mechanism of this class of numerical methods. Furthermore, through practical sessions on computers,
they will be able to put this theory into practice and run simulations on some examples.
Important information for doctoral students.
These lectures are part of the training program of the doctoral school EDSFA: the inscriptions are possible via ADUM.
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Planning
Class 1: Friday 27/01, 14h-16h, salle de conférences,
recorded session (Passcode: A*m5^!Fa)
Class 2: Tuesday 31/01 Friday 03/02 , 14h-16h, salle 1
Salle de conférences
, recorded session (Passcode: 1tH^2bGk)
Class 3: Monday 06/02, 14h-16h, salle 2,
recorded session (Passcode: 07q@?S2&)
Class 4: Wednesday 08/02, 14h-16h, salle de conférences. ,
recorded session (Passcode: #q+GQ4%M)
Class 5: Friday 10/02, 14h-16h, salle de conférences. Please come with your own laptop.
Zoom link, in case of problems, here is the
complete Zoom invitation.
Instructions for preparing the practical session, please have a look before the class: gitlab link
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Detailed outline
Part I : Origin of the MHM Method (class 1)
Part II : MHM's Basic Theory (classes 2, 3 and 4)
Part III : Practical Aspects of the Implementation of the MHM Method (classes 4 and 5)
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Shared documents
Here, you will find the shared folder with all the documents linked to the lectures.
Prerequisites
Basics of Finite Element methods and the theory of Partial Differential Equations.
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