Analysis and design of optimal discontinuous finite element schemes
Abstract/Contents
- Abstract
- Discontinuous finite-element schemes have the potential to provide highly accurate solutions to differential equations arising in a wide variety of applications ranging across mechanics, electro-magnetics, molecular chemistry, quantitative finance etc. Additionally, their design is intrinsically parallel, facilitating efficient acceleration through Graphical Processing Units. However, the complexity in formulation makes it very challenging to rigorously quantify the accuracy and stability offered by these schemes even for simple linear problems in one-dimension. In this dissertation, a rigorous framework is proposed that utilizes eigensolution analysis to develop consistency and convergence theory in the context of linear, hyperbolic conservation laws. It is proven that the general Flux Reconstruction (FR) formulation is consistent and converges at a time-dependent rate starting from a short-time value associated with polynomial interpolation and eventually asymptoting to a long-time value associated with dispersion and dissipation. This has led to the identification of a new class of schemes, designated Super-convergent FR (SFR), that exhibits an enhanced rate of convergence. Similarly, analysis of error for under-resolved simulations has led to the identification of a new set of high-order schemes, designated Optimal FR (OFR), that minimizes wave propagation errors for the range of resolvable wavenumbers. Finally, a computationally efficient, spectral filtering operation is proposed to stabilize solutions to non-linear, hyperbolic conservation laws. This has enabled accurate simulations of flows involving shock discontinuities even for extremely high polynomial orders. The filtered solution provides reduced total variation, reduced maximum overshoot/undershoot, and allows sub-element shocks to be localized in the interior of an element.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2016 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Asthana, Kartikey |
|---|---|
| Associated with | Stanford University, Department of Aeronautics and Astronautics. |
| Primary advisor | Jameson, Antony, 1934- |
| Thesis advisor | Jameson, Antony, 1934- |
| Thesis advisor | Lele, Sanjiva K. (Sanjiva Keshava), 1958- |
| Thesis advisor | Pinsky, P |
| Advisor | Lele, Sanjiva K. (Sanjiva Keshava), 1958- |
| Advisor | Pinsky, P |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Kartikey Asthana. |
|---|---|
| Note | Submitted to the Department of Aeronautics and Astronautics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2016. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2016 by Kartikey Asthana
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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