In bounded, polygonal domains $Ω\subset \mathbb{R}^2$ with Lipschitz boundary $\partialΩ$ consisting of a finite number of Jordan curves admitting analytic parametrizations, we analyze $hp$-FEM discretizations of linear, second order, singularly perturbed reaction diffusion equations on so-called geometric boundary layer meshes. We prove, under suitable analyticity assumptions on the data, that these $hp$-FEM afford exponential convergence in the natural "energy" norm of the problem, as long as the geometric boundary layer mesh can resolve the smallest length scale present in the problem. Numerical experiments confirm the robust exponential convergence of the proposed $hp$-FEM.
The main drawback for the application of the conforming Argyris FEM is the labourious implementation on the one hand and the low convergence rates on the other. If no appropriate adaptive meshes are utilised, only the convergence rate caused by corner singularities [Blum and Rannacher, 1980], far below the approximation order for smooth functions, can be achieved. The fine approximation with the Argyris FEM produces high-dimensional linear systems and for a long time an optimal preconditioned scheme was not available for unstructured grids. This paper presents numerical benchmarks to confirm t…
We report in this paper the analysis for the linear and nonlinear version of the flux corrected transport (FEM-FCT) scheme in combination with the backward Euler time-stepping scheme applied to time-dependent convection-diffusion-reaction problems. We present the stability and error estimates for the linear and nonlinear FEM-FCT scheme. Numerical results confirm the theoretical predictions.
We present the analysis for an $hp$ weak Galerkin-FEM for singularly perturbed reaction-convection-diffusion problems in one-dimension. Under the analyticity of the data assumption, we establish robust exponential convergence, when the error is measured in the energy norm, as the degree $p$ of the approximating polynomials is increased. The Spectral Boundary Layer mesh is used, which is the minimal (layer adapted) mesh for such problems. Numerical examples illustrating the theory are also presented.
Heriot-Watt University（赫瑞-瓦特大学）；Technische Universität Wien（维也纳工业大学）；University of Maryland（马里兰大学）；Universidad Técnica Federico Santa María（费德里科·圣塔玛利亚技术大学）；University of Tennessee（田纳西大学）；ETH Zürich（苏黎世联邦理工学院）
Certified Reachable Sets for Nonlinear Reaction--Diffusion Systems
University Mohammed VI Polytechnic（穆罕默德六世polytechnic大学）；Univ. Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，格勒诺布尔国立理工学院，GIPSA实验室）；De Vinci Higher Education, De Vinci Research Center（德芬奇高等教育，德芬奇研究中心）
For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains $D$, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete $hp$-time-stepping approach. We combine this se…
In nodal based finite element method (FEM), degrees of freedom are associated with the nodes of the element whereas, for edge FEM, degrees of freedom are assigned to the edges of the element. Edge element is constructed based on Whitney spaces. Nodal elements impose both tangential and normal continuity of vector or scalar fields across interface boundaries. But in edge elements only tangential continuity is imposed across interface boundaries, which is consistent with electromagnetic field problems. Therefore, the required continuities in the electromagnetic analysis are directly obtained wit…
Galerkin and Petrov-Galerkin projection-based reduced-order models (ROMs) of transient partial differential equations are typically obtained by performing a dimension reduction and projection process that is defined at either the spatially continuous or spatially discrete level. In both cases, it is common to add stabilization to the resulting ROM to increase the stability and accuracy of the method; the addition of stabilization is particularly common for advection-dominated systems when the ROM is under-resolved. While these two approaches can be equivalent in certain settings, differing tec…
