89 research outputs found
Numerical investigation of a 3D hybrid high-order method for the indefinite time-harmonic Maxwell problem
Hybrid High-Order (HHO) methods are a recently developed class of methods belonging to
the broader family of Discontinuous Sketetal methods. Other well known members of the
same family are the well-established Hybridizable Discontinuous Galerkin (HDG) method,
the nonconforming Virtual Element Method (ncVEM) and the Weak Galerkin (WG) method.
HHO provides various valuable assets such as simple construction, support for fully-polyhedral
meshes and arbitrary polynomial order, great computational efficiency, physical accuracy and
straightforward support for hp-refinement. In this work we propose an HHO method for the
indefinite time-harmonic Maxwell problem and we evaluate its numerical performance. In
addition, we present the validation of the method in two different settings: a resonant cavity
with Dirichlet conditions and a parallel plate waveguide problem with a total/scattered field
decomposition and a plane-wave boundary condition. Finally, as a realistic application, we
demonstrate HHO used on the study of the return loss in a waveguide mode converter
On slope limiting and deep learning techniques for the numerical solution to convection-dominated convection-diffusion problems
As the first main topic, several slope-limiting techniques from the literature are presented, and various novel methods are proposed. These post-processing techniques aim to automatically detect regions where the discrete solution has unphysical values and approximate the solution locally by a lower degree polynomial. This thesis's first major contribution is that two novel methods can reduce the spurious oscillations significantly and better than the previously known methods while preserving the mass locally, as seen in two benchmark problems with two different diffusion coefficients.
The second focus is showing how to incorporate techniques from machine learning into the framework of classical finite element methods. Hence, another significant contribution of this thesis is the construction of a machine learning-based slope limiter. It is trained with data from a lower-order DG method from a particular problem and applied to a higher-order DG method for the same and a different problem. It reduces the oscillations significantly compared to the standard DG method but is slightly worse than the classical limiters.
The third main contribution is related to physics-informed neural networks (PINNs) to approximate the solution to the model problem. Various ways to incorporate the Dirichlet boundary data, several loss functionals that are novel in the context of PINNs, and variational PINNs are presented for convection-diffusion-reaction problems. They are tested and compared numerically. The novel loss functionals improve the error compared to the vanilla PINN approach. It is observed that the approximations are free of oscillations and can cope with interior layers but have problems capturing boundary layers
Anisotropic analysis of VEM for time-harmonic Maxwell equations in inhomogeneous media with low regularity
It has been extensively studied in the literature that solving Maxwell
equations is very sensitive to the mesh structure, space conformity and
solution regularity. Roughly speaking, for almost all the methods in the
literature, optimal convergence for low-regularity solutions heavily relies on
conforming spaces and highly-regular simplicial meshes. This can be a
significant limitation for many popular methods based on polytopal meshes in
the case of inhomogeneous media, as the discontinuity of electromagnetic
parameters can lead to quite low regularity of solutions near media interfaces,
and potentially worsened by geometric singularities, making many popular
methods based on broken spaces, non-conforming or polytopal meshes particularly
challenging to apply. In this article, we present a virtual element method for
solving an indefinite time-harmonic Maxwell equation in 2D inhomogeneous media
with quite arbitrary polytopal meshes, and the media interface is allowed to
have geometric singularity to cause low regularity. There are two key
novelties: (i) the proposed method is theoretically guaranteed to achieve
robust optimal convergence for solutions with merely
regularity, ; (ii) the polytopal element shape can be highly
anisotropic and shrinking, and an explicit formula is established to describe
the relationship between the shape regularity and solution regularity.
Extensive numerical experiments will be given to demonstrate the effectiveness
of the proposed method
Numerical solution of the Biot/elasticity interface problem using virtual element methods
We propose, analyze and implement a virtual element discretization for an
interfacial poroelasticity-elasticity consolidation problem. The formulation of
the time-dependent poroelasticity equations uses displacement, fluid pressure,
and total pressure, and the elasticity equations are written in the
displacement-pressure formulation. The construction of the virtual element
scheme does not require Lagrange multipliers to impose the transmission
conditions (continuity of displacement and total traction, and no-flux for the
fluid) on the interface. We show the stability and convergence of the virtual
element method for different polynomial degrees, and the error bounds are
robust with respect to delicate model parameters (such as Lame constants,
permeability, and storativity coefficient). Finally, we provide numerical
examples that illustrate the properties of the scheme
Superconvergent P1 honeycomb virtual elements and lifted P3 solutions
When solving the Poisson equation on honeycomb hexagonal grids, we show that
the virtual element is three-order superconvergent in -norm, and
two-order superconvergent in and norms. We define a local
post-process which lifts the superconvergent solution to a solution
of the optimal-order approximation. The theory is confirmed by a numerical
test
A Locking-Free Weak Galerkin Finite Element Method for Linear Elasticity Problems
In this paper, we introduce and analyze a lowest-order locking-free weak
Galerkin (WG) finite element scheme for the grad-div formulation of linear
elasticity problems. The scheme uses linear functions in the interior of mesh
elements and constants on edges (2D) or faces (3D), respectively, to
approximate the displacement. An -conforming displacement
reconstruction operator is employed to modify test functions in the right-hand
side of the discrete form, in order to eliminate the dependence of the
parameter in error estimates, i.e., making the scheme
locking-free. The method works without requiring to be bounded. We prove optimal error estimates, independent of
, in both the -norm and the -norm. Numerical experiments
validate that the method is effective and locking-free
Mixed finite element approximation of porous media flows
The reliable simulation of flow in fractured porous media is a key aspect in the decision making process of stakeholders within politics and the geosciences, for example when assessing the suitability of burial sites for storage of high–level radioactive waste. This thesis aims to tackle the challenge that is the accurate simulation of these flows and does so via three computational developments. That is, suitable models for porous media flow with fractures; obtaining rigorous and reliable estimates of errors generated through these models; and the accurate simulation of the times–of–flight for particles transported by groundwater within the porous medium. Firstly, an expansion procedure for fractures in porous media is developed so that physical fluid laws are still retained when tracking particles across fracture–bulk interfaces. Moreover, the second contribution of this work is the utilisation of the dual–weighted–residual method to define suitable elementwise indicators for generic quantities of interest. The third contribution of this thesis is the attainment of accurate simulations of travel times for particles in porous media, achieved through linearising the functional representing the time–of–flight; in practice, numerical examples, including one inspired by the Sellafield site in Cumbria, UK, validate the performance of the proposed error estimator, and hence are useful in the safety assessment of storage facilities intended for radioactive waste
A space-time discontinuous Galerkin method for coupled poroelasticity-elasticity problems
This work is concerned with the analysis of a space-time finite element
discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical
discretization of wave propagation in coupled poroelastic-elastic media. The
mathematical model consists of the low-frequency Biot's equations in the
poroelastic medium and the elastodynamics equation for the elastic one. To
realize the coupling, suitable transmission conditions on the interface between
the two domains are (weakly) embedded in the formulation. The proposed PolydG
discretization in space is then coupled with a dG time integration scheme,
resulting in a full space-time dG discretization. We present the stability
analysis for both the continuous and the semidiscrete formulations, and we
derive error estimates for the semidiscrete formulation in a suitable energy
norm. The method is applied to a wide set of numerical test cases to verify the
theoretical bounds. Examples of physical interest are also presented to
investigate the capability of the proposed method in relevant geophysical
scenarios
Stabilization-free HHO a posteriori error control
The known a posteriori error analysis of hybrid high-order methods treats the stabilization contribution as part of the error and as part of the error estimator for an efficient and reliable error control. This paper circumvents the stabilization contribution on simplicial meshes and arrives at a stabilization-free error analysis with an explicit residual-based a posteriori error estimator for adaptive mesh-refining as well as an equilibrium-based guaranteed upper error bound (GUB). Numerical evidence in a Poisson model problem supports that the GUB leads to realistic upper bounds for the displacement error in the piecewise energy norm. The adaptive mesh-refining algorithm associated to the explicit residual-based a posteriori error estimator recovers the optimal convergence rates in computational benchmarks
Pressure-robust enriched Galerkin methods for the Stokes equations
In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for
solving the Stokes equations, which is an enhanced version of the EG scheme for
the Stokes problem proposed in [Son-Young Yi, Xiaozhe Hu, Sanghyun Lee, James
H. Adler, An enriched Galerkin method for the Stokes equations, Computers and
Mathematics with Applications, accepted, 2022]. The pressure-robustness is
achieved by employing a velocity reconstruction operator on the load vector on
the right-hand side of the discrete system. An a priori error analysis proves
that the velocity error is independent of the pressure and viscosity. We also
propose and analyze a perturbed version of our pressure-robust EG method that
allows for the elimination of the degrees of freedom corresponding to the
discontinuous component of the velocity vector via static condensation. The
resulting method can be viewed as a stabilized -conforming
- method. Further, we consider efficient block
preconditioners whose performances are independent of the viscosity. The
theoretical results are confirmed through various numerical experiments in two
and three dimensions
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