Title: Finite element methods for non H1 space very weak solutions. Part II. L2 projected
Lagrange continuous finite element methods for Maxwell’s equations
Speaker: Professor Huoyuan Duan (Nankai University, Tianjin, China)
Abstract: In this talk I will report my recent work on L2 projected Lagrange continuous finite
element method for Maxwell’s equations, with emphasis on a non H1 space solution. Since
1950s Lagrange continuous finite element methods in H1 space have been reigning over the
field of numerical solution of partial differential equations, they had never been really
successful for Maxwell’s equations with a non H1 space solution until the last ten years. Such an
embarrassing situation had been haunting both mathematical and engineering communities for
so long that it had been widely believed that Lagrange continuous finite element method would
have died for Maxwell’s equations. A breakthrough and rehabilitation took place when
approaching the beginning of the twenty century, labeled as the weighted regularization
method, by Costabel and Dauge, with a publication in Numerische Mathematik in 2002. Since
then, in 2004, a minus one Lagrange continuous finite element method by Bramble and
Pascicak was seen in Mathematics of Computation, and in 2007, 2009, our L2 projected
Lagrange continuous finite element method were published in SIAM Journal on Numerical
Analysis. In addition, a variant of weighted method in the context of dual least-squares method
is proposed by Manteuffel etc (2007， SIAM Journal on Numerical Analysis), a variant of
weighted in a mixed form is studied by Ciarlet Jr etc (2010, Numerische Mathematik), and a
variant of minus one method is analyzed by Guermond etc (2012, Mathematics of
Computation). The typical reason for a non H1 space solution is due to the irregular boundary
of the domain, i.e., there are reentrant corners and edges. An intuitive look at the non H1 space
solution is that the solution exhibits an infinity gradient locally, e.g., in the immediately vicinity
of some point in the domain. In my talk, I will report how our L2 projected Lagrange continuous
finite element method works for a non H1 space solution. Numerical tests in 2D and 3D are also
presented. A prominently significance in L2 projected Lagrange continuous finite element
method is that one never needs to care where and when and how the singularity of the solution
arises.
Acknowledgements: This speaker is partially supported by the National Natural Science
Foundation of China under the grants 11071132 and 11171168 and the Research Fund for the
Doctoral Program of Higher Education of China under the grant 20100031110002. This speaker
would like to thank Professor Roger C. E. Tan, for his support during the speaker’s visiting
National University of Singapore. The speaker’s thanks also go to Professor Weizhu Bao and
Professor Weiqing Ren for their kind invitation to give this talk at Department of Mathematics
of National University of Singapore.