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Full-Text Articles in Physical Sciences and Mathematics
The Dpg-Star Method, Leszek Demkowicz, Jay Gopalakrishnan, Brendan Keith
The Dpg-Star Method, Leszek Demkowicz, Jay Gopalakrishnan, Brendan Keith
Portland Institute for Computational Science Publications
This article introduces the DPG-star (from now on, denoted DPG*) finite element method. It is a method that is in some sense dual to the discontinuous Petrov– Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG* methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to othermethods in the literature round out the newly …
Space-Time Discretizations Using Constrained First-Order System Least Squares (Cfosls), Kirill Voronin, Chak Shing Lee, Martin Neumüller, Paulina Sepulveda, Panayot S. Vassilevski
Space-Time Discretizations Using Constrained First-Order System Least Squares (Cfosls), Kirill Voronin, Chak Shing Lee, Martin Neumüller, Paulina Sepulveda, Panayot S. Vassilevski
Portland Institute for Computational Science Publications
This paper studies finite element discretizations for three types of time-dependent PDEs, namely heat equation, scalar conservation law and wave equation, which we reformulate as first order systems in a least-squares setting subject to a space-time conservation constraint (coming from the original PDE). Available piece- wise polynomial finite element spaces in (n + 1)-dimensions for functional spaces from the (n + 1)-dimensional de Rham sequence for n = 3, 4 are used for the implementation of the method. Computational results illustrating the error behavior, iteration counts and performance of block-diagonal and monolithic geometric multi- grid preconditioners are …
The Auxiliary Space Preconditioner For The De Rham Complex, Jay Gopalakrishnan, Martin Neumüller, Panayot S. Vassilevski
The Auxiliary Space Preconditioner For The De Rham Complex, Jay Gopalakrishnan, Martin Neumüller, Panayot S. Vassilevski
Portland Institute for Computational Science Publications
We generalize the construction and analysis of auxiliary space preconditioners to the n-dimensional finite element subcomplex of the de Rham complex. These preconditioners are based on a generalization of a decomposition of Sobolev space functions into a regular part and a potential. A discrete version is easily established using the tools of finite element exterior calculus. We then discuss the four-dimensional de Rham complex in detail. By identifying forms in four dimensions (4D) with simple proxies, form operations are written out in terms of familiar algebraic operations on matrices, vectors, and scalars. This provides the basis for our implementation of …
A Parallel Mesh Generator In 3d/4d, Kirill Voronin
A Parallel Mesh Generator In 3d/4d, Kirill Voronin
Portland Institute for Computational Science Publications
In the report a parallel mesh generator in 3d/4d is presented. The mesh generator was developed as a part of the research project on space-time discretizations for partial differential equations in the least-squares setting. The generator is capable of constructing meshes for space-time cylinders built on an arbitrary 3d space mesh in parallel. The parallel implementation was created in the form of an extension of the finite element software MFEM. The code is publicly available in the Github repository