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The focus of this work is to investigate and to apply, for the first time, a very high-resolution correction procedure via reconstruction (CPR) numerical discretization technique for the hyperbolic saturation equation that describes 1-D oil-water displacement through heterogeneous porous media. The CPR method can achieve high-order accuracy via a compact stencil consisting of the current cell and its immediate neighbors; in addition, the CPR recovers simplified versions of nodal discontinuous Galerkin (NDG), spectral volume (SV), and spectral difference (SD) methods using an adequate polynomial reconstruction function, whose coefficients are preprocessed and stored. Indeed the CPR versions of NDG and SV/SD are highly efficient. In order to suppress numerical oscillations near shocks, which are typical from higher-order schemes, a hierarchical MLP (multi-dimensional limiting process) is used in the reconstruction stage. The integration in time is carried out using a third-order RK (Runge-Kutta) method. A number of 1-D two-phase flow benchmark problems are solved, to verify the accuracy, efficiency, and shock-capturing capability of the adopted methodology.

We design and analyze a hybrid high-order (HHO) method on unfitted meshes to approximate elliptic interface problems. The curved interface can cut through the mesh cells in a very general fashion. As in classical HHO methods, the present unfitted method introduces cell and face unknowns in uncut cells but doubles the unknowns in the cut cells and on the cut faces. The main difference with classical HHO methods is that a Nitsche-type formulation is used to devise the local reconstruction operator. As in classical HHO methods, cell unknowns can be eliminated locally leading to a global problem coupling only the face unknowns by means of a compact stencil. We prove stability estimates and optimal error estimates in the H¹-norm. Robustness with respect to cuts is achieved by a local cell-agglomeration procedure taking full advantage of the fact that HHO methods support polyhedral meshes. Robustness with respect to the contrast in the material properties from both sides of the interface is achieved by using material-dependent weights in Nitsche's formulation.

We present a new high-order finite element method for the discretization of partial differential equations on stationary smooth surfaces which are implicitly described as the zero level of a level set function. The discretization is based on a trace finite element technique. The higher discretization accuracy is obtained by using an isoparametric mapping of the volume mesh, based on the level set function, as introduced in [C. Lehrenfeld, Comp. Meth. Appl. Mech. Engrg., 300 (2016), pp. 716-733]. The resulting trace finite element method is easy to implement. We present an error analysis of this method and derive optimal order H¹(Г)-norm error bounds. A second topic of this paper is a unified analysis of several stabilization methods for trace finite element methods. Only a stabilization method which is based on adding an anisotropic diffusion in the volume mesh is able to control the condition number of the stiffness matrix also for the case of higher-order discretizations. Results of numerical experiments are included which confirm the theoretical findings on optimal order discretization errors and uniformly bounded condition numbers.

The travel time computation of microseismic waves in different directions (particularly, the diagonal direction) in three-dimensional space has been found to be inaccurate, which seriously affects the localization accuracy of three-dimensional microseismic sources. In order to solve this problem, this research study developed a method of calculating the P-wave travel time based on a 3D high-order fast marching method (3D_H_FMM). This study focused on designing a high-order finite-difference operator in order to realize the accurate calculation of the P-wave travel time in three-dimensional space. The method was validated using homogeneous velocity models and inhomogeneous layered media velocity models of different scales. The results showed that the overall mean absolute error (MAE) of the two homogenous models using 3D_H_FMM had been reduced by 88.335%, and 90.593% compared with the traditional 3D_FMM. On that basis, the three-dimensional localization of microseismic sources was carried out using a particle swarm optimization algorithm. The developed 3D_H_FMM was used to calculate the travel time, then to conduct the localization of the microseismic source in inhomogeneous models. The mean error of the localization results of the different positions in the three-dimensional space was determined to be 1.901 m, and the localization accuracy was found to be superior to that of the traditional 3D_FMM method (mean absolute localization error: 3.447 m) with the small-scaled inhomogeneous model.

The scattering of electromagnetic radiation by a layered periodic diffraction grating is an important model in engineering and the sciences. The numerical simulation of this experiment has been widely explored in the literature and we advocate for a novel interfacial method which is perturbative in nature. More specifically, we extend a recently developed High-Order Perturbation of Surfaces/Asymptotic Waveform Evaluation algorithm to utilize a stabilized numerical scheme which also suggests a rigorous convergence result. An implementation of this algorithm is described, validated, and utilized in a sequence of challenging and physically relevant numerical experiments.

We design and analyze an approximation method for advection-diffusion-reaction equations where the (generalized) degrees of freedom are polynomials of order k≥0 at mesh faces. The method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions. Fairly general meshes with polytopal and nonmatching cells are supported. Arbitrary polynomial orders can be considered, including the case k=0, which is closely related to mimetic finite difference/mixed-hybrid finite volume methods. The error analysis covers the full range of Péclet numbers, including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain. Computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications. Numerical results are presented.

In some previous works, the authors have introduced a strategy to develop well-balanced high-order numerical methods for nonconservative hyperbolic systems in the framework of path-conservative numerical methods. The key ingredient of these methods is a well-balanced reconstruction operator, i.e. an operator that preserves the stationary solutions in some sense. A strategy has been also introduced to modify any standard reconstruction operator like MUSCL, ENO, CWENO, etc. in order to be well-balanced. In this article, the specific case of 1d systems of balance laws is addressed and difficulties are gradually introduced: the methods are presented in the simpler case in which the source term does not involve Dirac masses. Next, systems whose source term involves the derivative of discontinuous functions are considered. In this case, the notion of weak solution is discussed and the Generalized Hydrostatic Reconstruction technique is used for the treatment of singular source terms. A technique to preserve the well-balancedness of the methods in the presence of numerical integration is introduced. The strategy is applied to derive first, second and third order well-balanced methods for Burgers’ equation with a nonlinear source term and for the Euler equations with gravity.

In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady nonlinear Leray–Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, LpL^{p}-stability and Ws,pW^{s,p}-approximation properties for L2L^{2}-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.

The plasmonics of two-dimensional materials, such as graphene, has become an important field of study for devices operating in the terahertz to midinfrared regime where such phenomena are supported. The semimetallic character of these materials permits electrostatic biasing which allows one to tune their electrical properties, unlike the noble metals (e.g., gold, silver) which also support plasmons. In the literature there are two principal approaches to modeling two-dimensional materials: With a thin layer of finite thickness featuring an effective permittivity, or with a surface current. We follow this latter approach to not only derive governing equations which are valid in the case of curved interfaces, but also reformulate these volumetric equations in terms of surface quantities using Dirichlet-Neumann operators. Such operators have been used extensively in the numerical simulation of electromagnetics problems, and we use them to restate the governing equations at layer interfaces. Beyond this, we show that these surface equations can be numerically simulated in an efficient, stable, and accurate fashion using a High-Order Perturbation of Surfaces methodology. We present detailed numerical results which not only validate our simulation using the Method of Manufactured Solutions and by comparison to results in the literature, but also describe Surface Plasmon Resonances at the \"wavy\" (corrugated) interface of a dielectric-graphene-dielectric structure.

We present the nonconforming virtual element method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable nonpolynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the nonpolynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two- and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the nonconforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.