# Abstracts

## S Brenner, Louisiana State University

## Finite Element Methods for Fourth Order Elliptic Variational Inequalities

Fourth order elliptic variational inequalities appear in obstacle problems for Kirchhoff plates and optimal control problems constrained by second order elliptic partial differential equations. The numerical analysis of these variational inequalities is more challenging than the analysis in the second order case because the complementarity forms of fourth order variational inequalities only exist in a weak sense. In this talk we will present a new approach to the analysis of finite element methods for fourth order elliptic variational inequalities that are applicable to C1 finite element methods, classical nonconforming finite element methods, and discontinuous Galerkin methods.

## D. Roy Mahapatra, Dept. of Aerospace Engg., IISc Bangalore

## Modelling enrichments in finite elements

This talk will cover three aspects of enrichment of finite elements for elliptic and hyperbolic PDEs arising in solid mechanics. First one is the field consistency based enrichment with an example of coupled fourth-order and second-order PDEs in context of vibration of beam. The second one is on the wavenumber-frequency space enrichment in context of elastic wave propagation problem. The third type of enrichment that will be discussed is on the geometric enrichment of finite element in context of material microstructure and fracture problem.

## Li-yeng Sung, Louisiana State University

## Multigrid Methods for Saddle Point Problems

In this talk I will present a general framework for the design and analysis of multigrid methods for saddle point problems arising from mixed finite element discretizations of elliptic boundary value problems. These multigrid methods are uniformly convergent in the energy norm on general polyhedral domains where the elliptic boundary value problems in general do not have full elliptic regularity. Applications to Stokes, Lam\'e, Darcy and related nonsymmetric systems will be discussed. This is joint work with Susanne Brenner, Hengguang Li and Duk-Soon Oh.

## Neela Nataraj, IIT Bombay

## Convergence of adaptive mixed finite element methods for general second order elliptic problems

In this talk, an adaptive mixed finite element method for second order elliptic problems defined on convex bounded polygonal domains is discussed and its convergence is analyzed. The main difficulties in the analysis are posed by the non-symmetric and indefinite form of the problem along with the lack of the orthogonality property in mixed finite element methods. The important tools in the analysis are a posteriori error estimators, quasi-orthogonality property and quasi-discrete reliability established using representation formula for the lowest-order Raviart-Thomas solution in terms of the Crouzeix-Raviart solution of the problem. An adaptive marking in each step for the local refinement is based on the edge residual and volume residual terms of the a posteriori estimator. Numerical experiments presented confirm the theoretical analysis. This is a joint work with Amiya K. Pani and Asha Dond Kisan.

## Tanmay Basak, Department of Chemical Engineering, IIT Madras

## Finite Element Based Simulation Studies on Various Heating Applications

Finite element method has been found to be a powerful technique for analyzing flow and thermal characteristics involving various heating applications such as natural and mixed convection and radiation assisted processing. Material processing and thermal management have received significant attention and in this context, this lecture is aimed to address thermal management for convection systems and various material processing scenarios during microwave assisted processing. Galerkin finite element method has been used to solve thermal and flow characteristics. Bi-quadratic elements are found to give converged solution with lesser mesh sizes compared to standard control volume or finite difference based solution strategies. Energy balance coupled with momentum balance equations are solved using Penalty finite element method. Finite element based analysis has been carried out for multiple solutions and bifurcations of steady natural convections in cavities. A generic spectrum of multiple symmetry breaking flow and thermal characteristics has been obtained based on robust finite element in house code. Further, thermal management has been analyzed via finite element post processing based on streamfunction, heatfunction, and entropy generation. Finite element method has been found to be extremely powerful for heat flow distributions via trajectories of heatlines (represented via heatfunctions). Major challenge on evaluation of heatfunctions is based on solutions of Poisson equations with integral boundary conditions within any irregular or non-square domain. Efficiency of a thermal convection system is quantified via estimation of exergy loss based on minimization of entropy generation due to heat transfer and fluid flow irreversibilities. Finite element based basis functions are also found to be robust on evaluation of non-linear derivatives involving both fluid flow and thermal irreversibilities especially in irregular domains. Case studies have been demonstrated for various domains involving triangular, trapezoidal and complex cavities with various heating sources at walls. A few case studies are also shown for natural convection involving various distributed heating patterns. Efficiency of processing and thermal management have been demonstrated for above cases via heatlines and entropy generation for various natural and mixed convection problems. Finite element simulations are also demonstrated for microwave assisted volumetric thermal processing for various samples. A major challenge on finite element modeling on microwave transport processes involves integro-differential boundary conditions with Helmholtz equations of complex electric field in irregular domains. A few interesting results based on resonances of power absorption and counter-intuitive thermal characteristics during microwave assisted thermal processing have been demonstrated.

## O. Pironneau, LJLL-UPMC

## Analysis of a Coupled Fluid-Structure Model with Applications

We propose and analyse a simplified fluid-structure coupled model for flows with compliant walls with application to arterial flow. As in [Nobile et al. 2002] the wall reaction to the fluid is modelled by a small displacements visco-elastic shell where the tangential stress components and displacements are neglected. We show that within this small displacement approximation a transpiration condition can be used which does not require an update of the geometry at each time step, for pipe flow at least. Such simplifications lead to a model which is well posed and for which a semi-implicit time discretisation can be shown to converge. We present some numerical results and a comparison with a standard test case taken from hemodynamics. The model is more stable and less computer demanding than full models with mesh motions. We apply the model to a 3D arterial flow with a stent. We shall also look at the inverse problem of finding the vessel's mechanical properties from the displacements.

## Harish Kumar, IIT Delhi

## Role of Riemann solvers in RKDG methods for MHD equations

It has been claimed that the particular numerical flux used in Runge–Kutta Discontinuous Galerkin (RKDG) methods does not have a significant effect on the results of high-order simulations. We investigate this claim for the case of compressible ideal magnetohydrodynamics (MHD). We also address the role of limiting in RKDG methods. For smooth nonlinear solutions, we find that the use of a more accurate Riemann solver in third-order simulations results in lower errors and more rapid convergence. However, in the corresponding fourth-order simulations we find that varying the Riemann solver has a negligible effect on the solutions. In the vicinity of discontinuities, we find that high-order RKDG methods behave in a similar manner to the second-order method due to the use of a piecewise linear limiter. Thus, for solutions dominated by discontinuities, the choice of Riemann solver in a high-order method has similar significance to that in a second-order method. Our analysis of second-order methods indicates that the choice of Riemann solver is highly significant, with the more accurate Riemann solvers having the lowest computational effort required to obtain a given accuracy. This allows the error in fourth-order simulations of a discontinuous solution to be mitigated through the use of a more accurate Riemann solver. Furthermore, we also look at the spectral performance of these schemes.

## Amiya K. Pani, IIT Bombay

## Superconvergent Results for a class of expanded DGM applied to second order elliptic boundary value problems

Since the analysis of nonlinear elliptic boundary value problem is through linearization, we, therefore, first discuss superconvergence results for nonselfadjoint linear elliptic problems using discontinuous Galerkin methods. The essential ingredient of the analysis is to construct an auxiliary projection which is based on the analysis of Cockburn et. al. [Math. Comp. 78(2009), pp. 1-24] for a selfadjoint linear elliptic equation. Then it is extended to derive superconvergence results for quasilinear elliptic problems. When piecewise polynomials of degree $k >1$ are used to approximate both the potential as well as the flux, it is shown that the error estimate for the discrete flux in $L^2$-norm is of order k + 1. On solving a discrete linear elliptic problem at each element, a suitable postprocessing of the discrete potential is developed and then, it is proved that the resulting post-processed potential converges with order of convergence $k + 2$ in $L^2$-norm. These results confirm superconvergent results for linear elliptic problems. (Joint work with Sangita Yadav and Eun-Jae Park)