Project Details
Abstract
1) Wider research context
Trefftz methods can be used to reduce the computational cost of numerical methods for PDEs by constructing special discrete spaces. In recent years, Trefftz basis functions have been used mainly in combination with discontinuous Galerkin (DG) methods. DG methods are finite element methods that allow for discontinuities in the discrete spaces of a finite element method, and are therefore well-suited to accommodate non-standard basis functions. Trefftz methods are typically limited to PDE problems with zero right hand side, constant coefficients, and usually require the explicit construction of Trefftz basis functions.
2) Objectives
We aim to expand the capabilities of the Trefftz-DG method, making it a versatile tool for reducing degrees of freedom in a wide range of numerical problems. Through this project, we will provide analytical tools for advanced applications and analysis of Trefftz methods. Our research will establish connections from Trefftz methods to adjacent topics, such as unfitted finite elements, problems in heterogeneous media, and fluid dynamic applications.
This project is purely focused on fundamental research into Trefftz methods, and not on industrial applications.
3) Methods
Key techniques for the derivation of new methods are the quasi-Trefftz and the embedded Trefftz method, which allow to remove restrictions of the standard Trefftz method and circumvent explicit construction of Trefftz basis functions. For the analysis, we will extend established tools from Trefftz methods to prove well-posedness for more complicated problems and non-zero right-hand-sides, and introduce new tools from mixed finite element methods to show optimal convergence rates.
4) Innovation
We analyze the first Trefftz-DG method for parabolic model problems and extend the quasi-Trefftz method to elliptic problems. For a novel embedded Trefftz method, we derive the numerical analysis also in a general elliptic setting.
We will also investigate further the combination of Trefftz methods and unfitted finite elements, where the geometry of an object is not captured by the mesh, but instead given implicitly. Trefftz methods present a unique opportunity to reduce degrees of freedom in unfitted DG methods, as other techniques, such as Hybrid-DG methods, do not apply easily. We will apply the Trefftz method to moving domains in space-time, to surface PDEs, and to coupled bulksurface problems. For surface and interface problems, we incorporating additional stability constraints directly into the Trefftz space further reducing the degrees of freedom.
Finally, a novel approach to fluid dynamics will be established by deriving a Trefftz method for the Stokes problem of different conformity. Furthermore, we will apply the embedded Trefftz method to nonlinear problems, such as the Navier-Stokes equations.
5) Primary researchers involved
Paul Stocker (PI), Ilaria Perugia (Mentor), Christoph Lehrenfeld
Trefftz methods can be used to reduce the computational cost of numerical methods for PDEs by constructing special discrete spaces. In recent years, Trefftz basis functions have been used mainly in combination with discontinuous Galerkin (DG) methods. DG methods are finite element methods that allow for discontinuities in the discrete spaces of a finite element method, and are therefore well-suited to accommodate non-standard basis functions. Trefftz methods are typically limited to PDE problems with zero right hand side, constant coefficients, and usually require the explicit construction of Trefftz basis functions.
2) Objectives
We aim to expand the capabilities of the Trefftz-DG method, making it a versatile tool for reducing degrees of freedom in a wide range of numerical problems. Through this project, we will provide analytical tools for advanced applications and analysis of Trefftz methods. Our research will establish connections from Trefftz methods to adjacent topics, such as unfitted finite elements, problems in heterogeneous media, and fluid dynamic applications.
This project is purely focused on fundamental research into Trefftz methods, and not on industrial applications.
3) Methods
Key techniques for the derivation of new methods are the quasi-Trefftz and the embedded Trefftz method, which allow to remove restrictions of the standard Trefftz method and circumvent explicit construction of Trefftz basis functions. For the analysis, we will extend established tools from Trefftz methods to prove well-posedness for more complicated problems and non-zero right-hand-sides, and introduce new tools from mixed finite element methods to show optimal convergence rates.
4) Innovation
We analyze the first Trefftz-DG method for parabolic model problems and extend the quasi-Trefftz method to elliptic problems. For a novel embedded Trefftz method, we derive the numerical analysis also in a general elliptic setting.
We will also investigate further the combination of Trefftz methods and unfitted finite elements, where the geometry of an object is not captured by the mesh, but instead given implicitly. Trefftz methods present a unique opportunity to reduce degrees of freedom in unfitted DG methods, as other techniques, such as Hybrid-DG methods, do not apply easily. We will apply the Trefftz method to moving domains in space-time, to surface PDEs, and to coupled bulksurface problems. For surface and interface problems, we incorporating additional stability constraints directly into the Trefftz space further reducing the degrees of freedom.
Finally, a novel approach to fluid dynamics will be established by deriving a Trefftz method for the Stokes problem of different conformity. Furthermore, we will apply the embedded Trefftz method to nonlinear problems, such as the Navier-Stokes equations.
5) Primary researchers involved
Paul Stocker (PI), Ilaria Perugia (Mentor), Christoph Lehrenfeld
| Short title | Anwendung und Analyse Trefftz-Methoden |
|---|---|
| Status | Active |
| Effective start/end date | 1/07/24 → 30/06/27 |
Keywords
- Trefftz methods
- discontinuous Galerkin
- unfitted finite elements
- fluid dynamics