Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA discontinuous Galerkin scheme for a large class of elliptic equations
234
0
0.0
(
0
)
Added by
Nils Leif Fischer
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We present a discontinuous Galerkin internal-penalty scheme that is applicable to a large class of linear and non-linear elliptic partial differential equations. The scheme constitutes the foundation of the elliptic solver for the SpECTRE numerical relativity code. As such it can accommodate (but is not limited to) elliptic problems in linear elasticity, general relativity and hydrodynamics, including problems formulated on a curved manifold. We provide practical instructions that make the scheme functional in a production code, such as instructions for imposing a range of boundary conditions, for implementing the scheme on curved and non-conforming meshes and for ensuring the scheme is compact and symmetric so it may be solved more efficiently. We report on the accuracy of the scheme for a suite of numerical test problems.
rate research
Read More
A considerable amount of attention has been given to discontinuous Galerkin methods for hyperbolic problems in numerical relativity, showing potential advantages of the methods in dealing with hydrodynamical shocks and other discontinuities. This paper investigates discontinuous Galerkin methods for the solution of elliptic problems in numerical relativity. We present a novel hp-adaptive numerical scheme for curvilinear and non-conforming meshes. It uses a multigrid preconditioner with a Chebyshev or Schwarz smoother to create a very scalable discontinuous Galerkin code on generic domains. The code employs compactification to move the outer boundary near spatial infinity. We explore the properties of the code on some test problems, including one mimicking Neutron stars with phase transitions. We also apply it to construct initial data for two or three black holes.
Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.
This paper proposes and analyzes an ultra-weak local discontinuous Galerkin scheme for one-dimensional nonlinear biharmonic Schr{o}dinger equations. We develop the paradigm of the local discontinuous Galerkin method by introducing the second-order spatial derivative as an auxiliary variable instead of the conventional first-order derivative. The proposed semi-discrete scheme preserves a few physically relevant properties such as the conservation of mass and the conservation of Hamiltonian accompanied by its stability for the targeted nonlinear biharmonic Schr{o}dinger equations. We also derive optimal $L^2$-error estimates of the scheme that measure both the solution and the auxiliary variable. Several numerical studies demonstrate and support our theoretical findings.
In this paper, we present a unified analysis of the superconvergence property for a large class of mixed discontinuous Galerkin methods. This analysis applies to both the Poisson equation and linear elasticity problems with symmetric stress formulations. Based on this result, some locally postprocess schemes are employed to improve the accuracy of displacement by order min(k+1, 2) if polynomials of degree k are employed for displacement. Some numerical experiments are carried out to validate the theoretical results.
In this paper, we develop a well-balanced oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. One notable feature of the constructed scheme is the well-balanced property, which preserves exactly the hydrostatic equilibrium solutions up to machine error. Another feature is the non-oscillatory property, which is very important in the numerical simulation when there exist some shock discontinuities. To control the spurious oscillations, we construct an OFDG method with an extra damping term to the existing well-balanced DG schemes proposed in [Y. Xing and C.-W. Shu, CICP, 1(2006), 100-134.]. With a careful construction of the damping term, the proposed method achieves both the well-balanced property and non-oscillatory property simultaneously without compromising any order of accuracy. We also present a detailed procedure for the construction and a theoretical analysis for the preservation of the well-balancedness property. Extensive numerical experiments including one- and two-dimensional space demonstrate that the proposed methods possess the desired properties without sacrificing any order of accuracy.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
