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Indian Institute of Science

1. Porwal, Kamana. A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities.

Degree: 2014, Indian Institute of Science

URL: http://hdl.handle.net/2005/3107

The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very pop-ular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different ele-ments. Moreover they are high order accurate and stable methods. Adaptive algorithms reﬁne the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh reﬁnement.
The solution of linear elliptic problem exhibits singularities due to change in boundary con-ditions, irregularity of coefﬁcients and reentrant corners in the domain. Apart from this, the solu-tion of variational inequality exhibits additional irregular behaviour due to occurrence of the free boundary (the part of the domain which is a priori unknown and must be found as a component of the solution). In the lack of full elliptic regularity of the solution, uniform reﬁnement is inefﬁcient and it does not yield optimal convergence rate. But adaptive reﬁnement, which is based on the residuals ( or a posteriori error estimator) of the problem, enhance the efﬁciency by reﬁning the mesh locally and provides the optimal convergence. In this thesis, we derive a posteriori error estimates of the DG methods for the elliptic variational inequalities of the ﬁrst kind and the second kind.
This thesis contains seven chapters including an introductory chapter and a concluding chap-ter. In the introductory chapter, we review some fundamental preliminary results which will be used in the subsequent analysis. In Chapter 2, a posteriori error estimates for a class of DG meth-ods have been derived for the second order elliptic obstacle problem, which is a prototype for elliptic variational inequalities of the ﬁrst kind. The analysis of Chapter 2 is carried out for the general obstacle function therefore the error estimator obtained therein involves the min/max func-tion and hence the computation of the error estimator becomes a bit complicated. With a mild assumption on the trace of the obstacle, we have derived a signiﬁcantly simple and easily com-putable error estimator in Chapter 3. Numerical experiments illustrates that this error estimator indeed behaves better than the error estimator derived in Chapter 2. In Chapter 4, we have carried out a posteriori analysis of DG methods for the Signorini problem which arises from the study of the frictionless contact problems. A nonlinear smoothing map from the DG ﬁnite element space to conforming ﬁnite element space has been constructed and used extensively, in the analysis of Chapter 2, Chapter 3 and Chapter 4. Also, a common property shared by all DG methods allows us to carry out the analysis in uniﬁed setting. In Chapter 5, we study the C0 interior penalty method for the plate frictional contact problem, which is a fourth order variational inequality…
*Advisors/Committee Members: Thirupathi, Gudi.*

Subjects/Keywords: Elliptical Variable Inequalities; Posteriori Analysis; Elliptic Variational Inequalities; Variational Inequalities; Discontinuous Galerkin Methods; Posteriori Error Control; Elliptic Obstacle Problem; Posteriori Error Analysis; Posteriori Error Estimator; Signorini Problem; Frictional Plate Contact Problem; Frictional Contact Problem; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Porwal, K. (2014). A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities. (Thesis). Indian Institute of Science. Retrieved from http://hdl.handle.net/2005/3107

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Porwal, Kamana. “A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities.” 2014. Thesis, Indian Institute of Science. Accessed September 19, 2019. http://hdl.handle.net/2005/3107.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Porwal, Kamana. “A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities.” 2014. Web. 19 Sep 2019.

Vancouver:

Porwal K. A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities. [Internet] [Thesis]. Indian Institute of Science; 2014. [cited 2019 Sep 19]. Available from: http://hdl.handle.net/2005/3107.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Porwal K. A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities. [Thesis]. Indian Institute of Science; 2014. Available from: http://hdl.handle.net/2005/3107

Not specified: Masters Thesis or Doctoral Dissertation

Indian Institute of Science

2. Srivastava, Shweta. Stabilization Schemes for Convection Dominated Scalar Problems with Different Time Discretizations in Time dependent Domains.

Degree: 2017, Indian Institute of Science

URL: http://etd.iisc.ernet.in/2005/3574 ; http://etd.iisc.ernet.in/abstracts/4443/G28424-Abs.pdf

Problems governed by partial differential equations (PDEs) in deformable domains, t Rd; d = 2; 3; are of fundamental importance in science and engineering. They are of particular relevance in the design of many engineering systems e.g., aircrafts and bridges as well as to the analysis of several biological phenomena e.g., blood ow in arteries. However, developing numerical scheme for such problems is still very challenging even when the deformation of the boundary of domain is prescribed a priori. Possibility of excessive mesh distortion is one of the major challenge when solving such problems with numerical methods using boundary tted meshes. The arbitrary Lagrangian- Eulerian (ALE) approach is a way to overcome this difficulty. Numerical simulations of convection-dominated problems have for long been the subject to many researchers. Galerkin formulations, which yield the best approximations for differential equations with high diffusivity, tend to induce spurious oscillations in the numerical solution of convection dominated equations. Though such spurious oscillations can be avoided by adaptive meshing, which is computationally very expensive on ne grids. Alternatively, stabilization methods can be used to suppress the spurious oscillations.
In this work, the considered equation is designed within the framework of ALE formulation. In the first part, Streamline Upwind Petrov-Galerkin (SUPG) finite element method with conservative ALE formulation is proposed. Further, the first order backward Euler and the second order Crank-Nicolson methods are used for the temporal discretization. It is shown that the stability of the semi-discrete (continuous in time) ALE-SUPG equation is independent of the mesh velocity, whereas the stability of the fully discrete problem is unconditionally stable for implicit Euler method and is only conditionally stable for Crank-Nicolson time discretization. Numerical results are presented to support the stability estimates and to show the influence of the SUPG stabilization parameter in a time-dependent domain.
In the second part of this work, SUPG stabilization method with non-conservative ALE formulation is proposed. The implicit Euler, Crank-Nicolson and backward difference methods are used for the temporal discretization. At the discrete level in time, the ALE map influences the stability of the corresponding discrete scheme with different time discretizations, and it leads to schemes where conservative and non-conservative formulations are no longer equivalent. The stability of the fully discrete scheme, irrespective of the temporal discretization, is only conditionally stable. It is observed from numerical results that the Crank-Nicolson scheme induces high oscillations in the numerical solution compare to the implicit Euler and the backward difference time discretiza-tions. Moreover, the backward difference scheme is more sensitive to the stabilization parameter k than the other time discretizations. Further, the difference between the solutions obtained with the conservative and…
*Advisors/Committee Members: Ganesan, Sashikumaar, Thirupathi, Gudi.*

Subjects/Keywords: Finite Elements Approximation; Convection-Diffusion-Reaction Equation; Convention Dominated Scalar Problem; Finite Element Methods; Arbitrary Lagrangian-Eulerian (ALE) Approach; Streamline Upwind Petrov-Galerkin (SUPG); Boundary And Interior Layers; ALE-SUPG Finite Element Method; Local Projection Stabilization (LPS); Discontinuous Galerkin (dG) in Time; Convection-diffusion Equations; Discontinuous Galerkin Method; Mathematics

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Srivastava, S. (2017). Stabilization Schemes for Convection Dominated Scalar Problems with Different Time Discretizations in Time dependent Domains. (Thesis). Indian Institute of Science. Retrieved from http://etd.iisc.ernet.in/2005/3574 ; http://etd.iisc.ernet.in/abstracts/4443/G28424-Abs.pdf

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Srivastava, Shweta. “Stabilization Schemes for Convection Dominated Scalar Problems with Different Time Discretizations in Time dependent Domains.” 2017. Thesis, Indian Institute of Science. Accessed September 19, 2019. http://etd.iisc.ernet.in/2005/3574 ; http://etd.iisc.ernet.in/abstracts/4443/G28424-Abs.pdf.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Srivastava, Shweta. “Stabilization Schemes for Convection Dominated Scalar Problems with Different Time Discretizations in Time dependent Domains.” 2017. Web. 19 Sep 2019.

Vancouver:

Srivastava S. Stabilization Schemes for Convection Dominated Scalar Problems with Different Time Discretizations in Time dependent Domains. [Internet] [Thesis]. Indian Institute of Science; 2017. [cited 2019 Sep 19]. Available from: http://etd.iisc.ernet.in/2005/3574 ; http://etd.iisc.ernet.in/abstracts/4443/G28424-Abs.pdf.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Srivastava S. Stabilization Schemes for Convection Dominated Scalar Problems with Different Time Discretizations in Time dependent Domains. [Thesis]. Indian Institute of Science; 2017. Available from: http://etd.iisc.ernet.in/2005/3574 ; http://etd.iisc.ernet.in/abstracts/4443/G28424-Abs.pdf

Not specified: Masters Thesis or Doctoral Dissertation