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We design a Hybrid High-Order method for elliptic problems on curved domains. The method uses a cut cell technique for the representation of the curved boundary and imposes Dirichlet boundary conditions using Nitsche’s method. The physical boundary can cut through the cells in a very general fashion and the method leads to optimal error estimates in the H¹-norm.
We propose a method to solve the acoustic wave equation on an immersed domain using the hybridizable discontinuous Galerkin method for spatial discretization and the arbitrary derivative method with local time stepping (LTS) for time integration. The method is based on a cut finite element approach of high order and uses level set functions to describe curved immersed interfaces. We study under which conditions and to what extent small time step sizes balance cut instabilities, which are present especially for high‐order spatial discretizations. This is done by analyzing eigenvalues and critical time steps for representative cuts. If small time steps cannot prevent cut instabilities, stabilization by means of cell agglomeration is applied and its effects are analyzed in combination with local time step sizes. Based on two examples with general cuts, performance gains of the LTS over the global time stepping are evaluated. We find that LTS combined with cell agglomeration is most robust and efficient.
We design and analyze a Hybrid High-Order 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 $H^1$-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 propose and validate a novel extension of Hybrid High-Order (HHO) methods to meshes featuring curved elements. HHO methods are based on discrete unknowns that are broken polynomials on the mesh and its skeleton. We propose here the use of physical frame polynomials over mesh elements and reference frame polynomials over mesh faces. With this choice, the degree of face unknowns must be suitably selected in order to recover on curved meshes the same convergence rates as on straight meshes. We provide an estimate of the optimal face polynomial degree depending on the element polynomial degree and on the so-called effective mapping order. The estimate is numerically validated through specifically crafted numerical tests. All test cases are conducted considering two- and three-dimensional pure diffusion problems, and include comparisons with discontinuous Galerkin discretizations. The extension to agglomerated meshes with curved boundaries is also considered.
We develop an arbitrary-order primal method for diffusion problems on general polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstone of the method is a local (elementwise) discrete gradient reconstruction operator. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. The scheme is proved to optimally converge in the energy norm and in the
We devise an arbitrary-order locking-free method for linear elasticity. The method relies on a pure-displacement (primal) formulation and leads to a symmetric, positive definite system matrix with compact stencil. The degrees of freedom are vector-valued polynomials of arbitrary order k⩾1k⩾1 on the mesh faces, so that in three space dimensions, the lowest-order scheme only requires 9 degrees of freedom per mesh face. The method can be deployed on general polyhedral meshes. The key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems. The discrete problem is assembled cell-wise using these operators and a high-order stabilization bilinear form. Locking-free error estimates are derived for the energy norm and for the L2L2-norm of the displacement, with optimal convergence rates of order (k+1)(k+1) and (k+2)(k+2), respectively, for smooth solutions on general meshes. The theoretical results are confirmed numerically, and the CPU cost is evaluated on both standard and polygonal meshes.
We present a discontinuous Galerkin method, based on the classical method of Nitsche, for elliptic problems with an immersed boundary representation on a structured grid. In such methods very small elements typically occur at the boundary, leading to breakdown of the discrete coercivity as well as numerical instabilities. In this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain. This construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates. Furthermore, we prove a bound of the condition number of the stiffness matrix. All the results are valid for polynomials of arbitrary order. We also discuss the implementation of the method and present numerical examples in three dimensions.
In this Note we discuss a simple penalty method that allows to increase the robustness of fictitious domain methods. In particular the condition number of the matrix can be upper bounded independently of how the domain boundary intersects the computational mesh, under rather weak assumptions.
This paper considers the finite-element approximation of the elliptic interface problem: -▽·(σ▽u) + cu = f in Ω ⊂ Rn (n = 2 or 3), with u = 0 on ∂Ω, where σ is discontinuous across a smooth surface Γ in the interior of Ω. First we show that, if the mesh is isoparametrically
fitted to Γ using simplicial elements of degree k - 1, with k ≥ 2, then the standard Galerkin method achieves the optimal rate of convergence in the H1 and L2 norms over the approximations Ωl4 of Ωl where Ω ≡ Ωl ∪ Γ ∪ Ω2. Second, since it may be computationally inconvenient to fit the mesh to Γ, we analyse a fully practical piecewise linear
approximation of a related penalized problem, as introduced by Babuska (1970), based on a mesh that is independent of Γ. We
show that, by choosing the penalty parameter appropriately, this approximation converges to u at the optimal rate in the H1 norm over Ωl4 and in the L2 norm over any interior domain Ωl* satisfying Ωl* Ωl** Ωl4 for some domain Ωl**.
In this paper, we analyze the error of a fictitious domain method with a Lagrange multiplier. It is applied to solve a non
homogeneous elliptic Dirichlet problem with conforming finite elements of degree one on a regular grid. The main point is
the proof of a uniform inf-sup condition that holds provided the step size of the mesh on the actual boundary is sufficiently
large compared to the size of the interior grid.
Dans cet article, nous étudions l’erreur d’une méthode de domaine fictif avec multiplicateur de Lagrange. Nous l’appliquons
à la résolution d’un problème elliptique avec condition de Dirichlet non-homogène au bord par une méthode d’éléments finis
conforme de degré un sur une grille uniforme. Ceci repose sur la démonstration d’une condition inf-sup uniforme qui est satisfaite
lorsque le pas de la discrétisation sur la frontière du domaine d’origine est suffisamment grand comparé au pas de la grille
In this paper we propose a method for the finite element solution of elliptic interface problem, using an approach due to Nitsche. The method allows for discontinuities, internal to the elements, in the approximation across the interface. We show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh. Further, we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples.