**Seminars and Colloquiums**

for the week of May 15, 2017

for the week of May 15, 2017

*SPEAKERS*

**Friday**

Speaker: Stefan Schnake, UTK

**Friday, May 19th**

##### PhD DISSERTATION DEFENSE

Title: Numerical Methods for Non-divergence Form Second Order Linear Elliptic Partial Differential Equations and Discontinuous Ritz Methods for Problems from the Calculus of Variations

Speaker: Stefan Schnake, UTK

Time: 9:30a - 10:30a

Room: Ayres 111

This defense is concerned with design, analysis and computer implementation of numerical methods for non-divergence form second order linear elliptic partial differential equations (PDEs) and the related fully nonlinear Hamilton-Jacobi-Bellman equations, and for problems from the calculus of variations. These PDEs and calculus of variations problems arise from various scientific and engineering applications such as differential geometry, image processing, mathematical finance, deterministic and stochastic optimal control, materials science, and nonlinear elasticity.

The defense consists of two integral parts. In part one, we study discontinuous Galerkin approximations of a class of non-divergence form second order linear elliptic PDEs whose coefficients are only continuous. We develop an interior penalty discontinuous Galerkin (IP-DG) method for this class of PDEs and show the stability of the method and error estimates in a discrete $W^{2,p}$-norm. We also study the convergence of the vanishing moment method for this class of PDEs. The vanishing moment method is technique by which we approximate these PDEs by a family of fourth order PDEs. We present uniform $H^1$ and $H^2$-stability estimates for the approximate solutions and their convergence.

In part two, we study numerical methods for problems from the calculus of variations. We first build finite element approximations of a class of calculus of variations problems, which exhibit the so-called Lavrentiev gap phenomenon (LGP). Such a phenomenon often indicates the solution of the problem has a singularity. The LGP incapacitates all standard numerical methods, especially the finite element method, as they fail to produce a correct approximate solution. To overcome the difficulty, we build an enhanced finite element method based on a truncation technique and show numerically that the proposed enhanced finite element method correctly computes the solutions of several benchmark problems with the LGP. We also develop a discontinuous Galerkin numerical framework for general calculus of variations problems, which is called the discontinuous Ritz (DR) methodology and can be regarded as the counterpart of the discontinuous Galerkin (DG) methodology for PDEs. Conceptually, it approximates the admissible space by the DG spaces which consist of totally discontinuous piecewise polynomials and approximates the underlying energy functional by discrete energy functionals defined on the DG spaces. The main idea here is to construct the desired discrete energy functional by using the newly developed DG finite element calculus theory, which only requires replacing the gradient operator in the energy functional by the corresponding DG finite element discrete gradient and adding the standard interior penalty term to weakly enforce continuity. We prove that for a class convex energy functionals the proposed DR method has a nice compactness property, which, in turn, deduces the $\Gamma$-convergence of the DR discrete energy functional. As a by-product, this DR methodology can also be enhanced by the above mentioned truncation technique to handle calculus of variations problems which exhibit the Lavrentiev gap phenomenon.

Committee Members: Xiaobing Feng (Chair), Ohannes Karakashian, Tuoc Phan, Stanimire Tomov (EECS)

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**Past notices:**

3/13/17 - Spring Break

Winter Break