This paper focuses on the numerical solution of elliptic partial differential equations (PDEs) with Dirichlet and mixed boundary conditions, specifically addressing the challenges arising from irregular domains. Both finite element method (FEM) and finite difference method (FDM), face difficulties in dealing with arbitrary domains. The paper introduces a novel nodal symmetric ghost {method based on a variational formulation}, which combines the advantages of FEM and FDM. The method employs bilinear finite elements on a structured mesh and provides a detailed implementation description. A rigorous a priori convergence rate analysis is also presented. The convergence rates are validated with many numerical experiments, in both one and two space dimensions.
This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed. Given explicitly estimated error bounds between an exact solution $ u $ and a numerically computed approximate solution $ \hat{u} $, we evaluate the number of sign-changes of $ u $ (the number of nodal domains) and determine the location of zero level-sets of $ u $ (the location of the nodal line). We apply this method to the Dirichlet problem of the Allen-Cahn…
In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM), and its local kernel variants, to approximate second-order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points, specified along the estimated tangential direction at the sampled points at the boundary. The resulting GPDM estimator restr…
A high-order nodally bound-preserving and mass-conservative method for linear fourth-order elliptic problems and its applications to nonlinear parabolic equations
We develop a stabilized cut discontinuous Galerkin framework for the numerical solution of el- liptic boundary value and interface problems on complicated domains. The domain of interest is embedded in a structured, unfitted background mesh in R d , so that the boundary or interface can cut through it in an arbitrary fashion. The method is based on an unfitted variant of the classical symmetric interior penalty method using piecewise discontinuous polynomials defined on the back- ground mesh. Instead of the cell agglomeration technique commonly used in previously introduced unfitted discontinu…
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…
Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator
Institute of Computer Sciences, Czech Academy of Science（捷克科学院计算机科学研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院）；Department of Mathematics, University of Oslo（奥斯陆大学数学系）；Department of Numerical Analysis and Scientific Computing, Simula Research Laboratory（Simula研究实验室数值分析与科学计算部）；Faculty of Science and Technology, Norwegian University of Life Sciences（挪威生命科技大学理学院）
In this paper we consider a class of fictitious domain finite element methods known from the literature. These methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson equation. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The finite element space is split into two subspaces, where one subspace is spanned by all nodal basis functions corresponding to nodes on the b…
Motivated by many applications in complex domains with boundaries exposed to large topological changes or deformations, fictitious domain methods regard the actual domain of interest as being embedded in a fixed Cartesian background. This is usually achieved via a geometric parameterization of its boundary via level-set functions. In this note, the a priori analysis of unfitted numerical schemes with cut elements is extended beyond the realm of linear problems. More precisely, we consider the discretization of semilinear elliptic boundary value problems of the form $- Δu +f_1(u)=f_2$ with poly…
Elliptic interface problems whose solutions are $C^0$ continuous have been well studied over the past two decades. The well-known numerical methods include the strongly stable generalized finite element method (SGFEM) and immersed FEM (IFEM). In this paper, we study numerically a larger class of elliptic interface problems where their solutions are discontinuous. A direct application of these existing methods fails immediately as the approximate solution is in a larger space that covers discontinuous functions. We propose a class of high-order enriched unfitted FEMs to solve these problems wit…
