89 research outputs found

    Numerical investigation of a 3D hybrid high-order method for the indefinite time-harmonic Maxwell problem

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    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

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    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

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    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 Hθ\mathbf{H}^{\theta} regularity, θ∈(1/2,1]\theta\in(1/2,1]; (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

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    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

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    When solving the Poisson equation on honeycomb hexagonal grids, we show that the P1P_1 virtual element is three-order superconvergent in H1H^1-norm, and two-order superconvergent in L2L^2 and L∞L^\infty norms. We define a local post-process which lifts the superconvergent P1P_1 solution to a P3P_3 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

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    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 H(div)H(div)-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 LameˊLam\acute{e} parameter λ\lambda in error estimates, i.e., making the scheme locking-free. The method works without requiring λ∥∇⋅u∥1\lambda \|\nabla\cdot \mathbf{u}\|_1 to be bounded. We prove optimal error estimates, independent of λ\lambda, in both the H1H^1-norm and the L2L^2-norm. Numerical experiments validate that the method is effective and locking-free

    Mixed finite element approximation of porous media flows

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    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

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    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

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    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

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    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 H1H^1-conforming P1\mathbb{P}_1-P0\mathbb{P}_0 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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