An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, Part II. Piecewise-smooth interfaces
We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the reliable cell merging algorithm for smooth interfaces to automatically generate the induced mesh for piecewise smooth interfaces. An $hp$ a posteriori error estimate is derived for a new unfitted finite element method whose finite element functions are conforming in each subdomain. Numerical examples illustrate the competitive performance of the method.
This paper presents a lowest-order immersed Raviart-Thomas mixed triangular finite element method for solving elliptic interface problems on unfitted meshes independent of the interface. In order to achieve the optimal convergence rates on unfitted meshes, an immersed finite element finite (IFE) is constructed by modifying the traditional Raviart-Thomas element. Some important properties are derived including the unisolvence of IFE basis functions, the optimal approximation capabilities of the IFE space and the corresponding commuting digram. Optimal error estimates are rigorously proved for t…
We propose a discontinuous Galerkin(DG) method to approximate the elliptic interface problem on unfitted mesh using a new approximation space. The approximation space is constructed by patch reconstruction with one degree of freedom per element. The optimal error estimates in both L2 norm and DG energy norm are obtained, without the typical constraints for DG method on how the interface intersects to the elements in the mesh. Other than enjoying the advantages of DG method, our method may achieve even better efficiency than the conforming finite element method, as illustrated by numerical exam…
Immersed finite element (IFE) methods are a group of long-existing numerical methods for solving interface problems on unfitted meshes. A core argument of the methods is to avoid mesh regeneration procedure when solving moving interface problems. Despite the various applications in moving interface problems, a complete theoretical study on the convergence behavior is still missing. This research is devoted to close the gap between numerical experiments and theory. We present the first fully discrete analysis including the stability and optimal error estimates for a backward Euler IFE method fo…
A virtual element method (VEM) with the first order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual H(curl)-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of th…
We present a method of CutFEM type for the Poisson problem with either Dirichlet or Neumann boundary conditions. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements intersecting the domain where the problem is posed. The resulting mesh does not thus fit the boundary of the problem domain. Several finite element methods (XFEM, CutFEM) adapted to such meshes have been recently proposed. The originality of the present article consists in avoiding integration over the elements cut by the boundary of the problem domain, while preser…
We design an adaptive unfitted finite element method on the Cartesian mesh with hanging nodes. We derive an hp-reliable and efficient residual type a posteriori error estimate on K-meshes. A key ingredient is a novel hp-domain inverse estimate which allows us to prove the stability of the finite element method under practical interface resolving mesh conditions and also prove the lower bound of the hp a posteriori error estimate. Numerical examples are included.
Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a regular background mesh, not fitting the domain and its real boundary exactly. Our technique follows the phi-FEM paradigm, which supposes that the domain is given by a level-set function. In this paper, we prove a priori error estimates in l2(H1) and linf(L2) norms for an implicit Euler discretization in time. We give numerical illustrations to highlight the performances of phi-FEM, which combines optimal convergence accuracy, easy impleme…
We present an isoparametric unfitted finite element approach of the CutFEM or Nitsche-XFEM family for the simulation of two-phase Stokes problems with slip between phases. For the unfitted generalized Taylor--Hood finite element pair $\mathbf{P}_{k+1}-P_k$, $k\ge1$, we show an inf-sup stability property with a stability constant that is independent of the viscosity ratio, slip coefficient, position of the interface with respect to the background mesh and, of course, mesh size. In addition, we prove stability and optimal error estimates that follow from this inf-sup property. We provide numeric…
An interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method of arbitrary order is proposed for linear elasticity interface problems on unfitted meshes with respect to the interface and domain boundary. The method uses piecewise polynomials of degrees $k\ (>= 1)$ and $k-1$ respectively for the displacement and stress approximations in the interior of elements inside the subdomains separated by the interface, and piecewise polynomials of degree $k$ for the numerical traces of the displacement on the inter-element boundaries inside the subdomains and on the interfac…
In this work, we investigate the performance CutFEM as a high fidelity solver as well as we construct a competent and economical reduced order solver for PDE-constrained optimization problems in parametrized domains that live in a fixed background geometry and mesh. Its effectiveness and reliability will be assessed through its application for the numerical solution of quadratic optimization problems with elliptic equations as constraints, examining an archetypal case. The reduction strategy will be via Proper Orthogonal Decomposition of suitable FE snapshots, using an aggregated state and adj…
In this article, we develop and analyze a finite element method with the first family Nédélec elements of the lowest degree for solving a Maxwell interface problem modeled by a $\mathbf{H}(\text{curl})$-elliptic equation on unfitted meshes. To capture the jump conditions optimally, we construct and use $\mathbf{H}(\text{curl})$ immersed finite element (IFE) functions on interface elements while keep using the standard Nédélec functions on all the non-interface elements. We establish a few important properties for the IFE functions including the unisolvence according to the edge degrees of free…
Preconditioned Reconstructed Discontinuous Approximation For Elliptic Interface Problem on Unfitted Mesh
Peking University（北京大学）；Hangzhou International Innovation Institute of Beihang University（北京航空航天大学杭州创新研究院）；Sichuan University（四川大学）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
University of Science and Technology of China（中国科学技术大学）；The Hong Kong Polytechnic University（香港理工大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
In this work, we analyze an unfitted discontinuous Galerkin discretization for the numerical solution of the Stokes system based on equal higher-order discontinuous velocities and pressures. This approach combines the best from both worlds, firstly the advantages of a piece-wise discontinuous high-order accurate approximation and secondly the advantages of an unfitted to the true geometry grid around possibly complex objects and/or geometrical deformations. Utilizing a fictitious domain framework, the physical domain of interest is embedded in an unfitted background mesh and the geometrically …
In this work, we consider the modeling of inclusions in the material using an unfitted finite element method. In the unfitted methods, structured background meshes are used and only the underlying finite element space is modified to incorporate the discontinuities, such as inclusions. Hence, the unfitted methods provide a more flexible framework for modeling the materials with multiple inclusions. We employ the method of Lagrange multipliers for enforcing the interface conditions between the inclusions and matrix, this gives rise to the linear system of equations of saddle point type. We utili…
A ϕ-FEM approach for time-dependent domains with unfitted meshes
Université de Strasbourg, CNRS, Inria（斯特拉斯堡大学，法国国家科学研究中心，法国信息与自动化研究所）；Université Marie et Louis Pasteur, CNRS（玛丽·路易·巴斯德大学，法国国家科学研究中心）；University of Göttingen（哥廷根大学）
Unfitted (also known as embedded or immersed) finite element approximations of partial differential equations are very attractive because they have much lower geometrical requirements than standard body-fitted formulations. These schemes do not require body-fitted unstructured mesh generation. In turn, the numerical integration becomes more involved, because one has to compute integrals on portions of cells (only the interior part). In practice, these methods are restricted to level-set (implicit) geometrical representations, which drastically limit their application. Complex geometries in ind…
In this work, we present an adaptive unfitted finite element scheme that combines the aggregated finite element method with parallel adaptive mesh refinement. We introduce a novel scalable distributed-memory implementation of the resulting scheme on locally-adapted Cartesian forest-of-trees meshes. We propose a two-step algorithm to construct the finite element space at hand by means of a discrete extension operator that carefully mixes aggregation constraints of problematic degrees of freedom, which get rid of the small cut cell problem, and standard hanging degree of freedom constraints, whi…
This paper proposes an interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method for Darcy-Stokes-Brinkman interface problems in two and three dimensions. The method uses piecewise linear polynomials for the velocity approximation and piecewise constants for both the velocity gradient and pressure approximations in the interior of elements inside the subdomains separated by the interface, uses piecewise constants for the numerical traces of velocity on the inter-element boundaries inside the subdomains, and uses piecewise constants or linear polynomials for the nu…
We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on …
In this paper, we propose an unfitted Nitsche's method to compute the band structures of phononic crystal with impurities of general geometry. The proposed method does not require the background mesh to fit the interfaces of impurities, and thus avoids the expensive cost of generating body-fitted meshes and simplifies the inclusion of interface conditions in the formulation. The quasi-periodic boundary conditions are handled by the Floquet-Bloch transform, which converts the computation of band structures into an eigenvalue problem with periodic boundary conditions. More importantly, we show t…
The paper develops an unfitted finite element method for solving the Darcy system of equations posed in a network of fractures embedded in a porous matrix. The approach builds on the Hughes--Masud stabilized formulation of the Darcy problem and the trace finite element method. The system of fractures is allowed to cut through the background mesh in an arbitrary way. Moreover, the fractures are not triangulated in the common sense and the junctions of fractures are not fitted by the mesh. To couple the flow variables at multiple fracture junctions, we extend the Hughes--Masud formulation by inc…
The paper addresses stability and finite element analysis of the stationary two-phase Stokes problem with a piecewise constant viscosity coefficient experiencing a jump across the interface between two fluid phases. We first prove a priori estimates for the individual terms of the Cauchy stress tensor with stability constants independent of the viscosity coefficient. Next, this stability result is extended to the approximation of the two-phase Stokes problem by a finite element method. In the method considered, the interface between the phases does not respect the underlying triangulation, whi…
In this work, we bridge standard adaptive mesh refinement and coarsening on scalable octree background meshes and robust unfitted finite element formulations for the automatic and efficient solution of large-scale nonlinear solid mechanics problems posed on complex geometries, as an alternative to standard body-fitted formulations, unstructured mesh generation and graph partitioning strategies. We pay special attention to those aspects requiring a specialized treatment in the extension of the unfitted h-adaptive aggregated finite element method on parallel tree-based adaptive meshes, recently …
We propose and analyze an unfitted finite element method for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. The boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. Optimal convergence rates under the $L^2$ norm and the energy norm are de…
We propose an unfitted finite element method for numerically solving the time-harmonic Maxwell equations on a smooth domain. The model problem involves a Lagrangian multiplier to relax the divergence constraint of the vector unknown. The embedded boundary of the domain is allowed to cut through the background mesh arbitrarily. The unfitted scheme is based on a mixed interior penalty formulation, where Nitsche penalty method is applied to enforce the boundary condition in a weak sense, and a penalty stabilization technique is adopted based on a local direct extension operator to ensure the stab…
We present a priori and a posteriori error analysis of a high order hybridizable discontinuous Galerkin (HDG) method applied to a semi-linear elliptic problem posed on a piecewise curved, non polygonal domain. We approximate $Ω$ by a polygonal subdomain $Ω_h$ and propose an HDG discretization, which is shown to be optimal under mild assumptions related to the non-linear source term and the distance between the boundaries of the polygonal subdomain $Ω_h$ and the true domain $Ω$. Moreover, a local non-linear post-processing of the scalar unknown is proposed and shown to provide an additional ord…
In this work, we analyse the links between ghost penalty stabilisation and aggregation-based discrete extension operators for the numerical approximation of elliptic partial differential equations on unfitted meshes. We explore the behavior of ghost penalty methods in the limit as the penalty parameter goes to infinity, which returns a strong version of these methods. We observe that these methods suffer locking in that limit. On the contrary, aggregated finite element spaces are locking-free because they can be expressed as an extension operator from well-posed to ill-posed degrees of freedom…
