A priori error analysis of the hp-Mortar FEM for elliptic and parabolic problems
| Speaker |
Dr. Sanjib Kumar Acharya,
The LNM Institute of Information Technology, Jaipur
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|---|---|
| When |
Aug 02, 2017
from 04:00 PM to 05:00 PM |
| Where | LH 006 |
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Abstract: In this talk, we start with the a priori error estimates for primal hybrid finite element methods. Then we discuss the a priori estimates for the hp-version of the mortar finite element methods for elliptic and parabolic problems problems with a general mesh condition. For parabolic problems, we establish the quasioptimal error estimates in L2 - and piecewise H1 -norms for the semidiscrete method. With an extra regularity assumption of the solution we establish the superconvergence estimates in negative norms. We introduce a Lagrange multiplier formulation and establish the error estimates for both primal and flux variables. We obtain exponential decay of the error (for both primal and flux variables) in the spatial direction when geometric meshes are employed over subdomains. We discuss a fully discrete scheme using backward Euler finite difference in the temporal direction and establish its error estimates in L2 - and piecewise H1 -norms.
To circumvent the requirement of inf-sup condition in mortar methods with a Lagrange multiplier, we introduce a hp-stabilized Nitsche’s mortar method. We discuss the stabilized method for both elliptic and parabolic initial boundary value problems. The inf-sup condition has been circumvented using a penalty term of O(h/p2 ) in the formulation. The stability of the method is established in a mesh dependent norm with the help of a hp-inverse inequality, which ensures the existence and uniqueness of the mortar solution. We obtain the piecewise H1 -error estimates for the primal variable and the estimates for the flux variable simultaneously with the help of mesh dependent norm. We obtain optimal error estimates with respect to h and suboptimal estimates with respect to p for both the variables. We also discuss both semidiscrete and fully discrete schemes for the parabolic problems and analyze the error estimates for both primal and flux variables. Finally, we give a summary and some possible future extensions.