Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the application of high-order Finite Element (FE) to certain classes of PDEs, such as diffusion equations and the Navier-Stokes (NS) equations, often leads to numerical instabilities. These instabilities limit the number of valid terms in the series, though the efficiency of time series integration even when resummation techniques like the Borel-Padé-Laplace (BPL) integrators are employed. In this study, we introduce a novel variational formulation for computing the terms of a TSE associated with a given PDE using higher-order FEs. Our approach involves the incorporation of artificial diffusion terms on the left-hand side of the equations corresponding to each power in the series, serving as a stabilization technique. We demonstrate that this method can be interpreted as a minimization of an energy functional, wherein the total variations of the unknowns are considered. Furthermore, we establish that the coefficients of the artificial diffusion for each term in the series obey a recurrence relation, which can be determined by minimizing the condition number of the associated linear system. We highlight the link between the proposed technique and the Discrete Maximum Principle (DMP) of the heat equation. We show, via numerical experiments, how the proposed technique allows having additional valid terms of the series that will be substantial in enlarging the stability domain of the BPL integrators.
In this paper, a stabilized second order in time accurate linear exponential time differencing (ETD) scheme for the no-slope-selection thin film growth model is presented. An artificial stabilizing term $Aτ^2\frac{\partialΔ^2 u}{\partial t}$ is added to the physical model to achieve energy stability, with ETD-based multi-step approximations and Fourier collocation spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long time energy stability and detailed $\ell^{\infty}(0,T; \ell^2)$ error analysis are provided based on the energy met…
A stabilized trace finite element method for partial differential equations on evolving surfaces
University of Göttingen（哥廷根大学）；University of Houston（休斯顿大学）；Sechenov University（谢切诺夫大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所，国家数学与交叉科学中心）
Stabilized Weighted Reduced Order Methods for Parametrized Advection-Dominated Optimal Control Problems governed by Partial Differential Equations with Random Inputs
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide a fully discrete strong convergence analysis of a family of explicit stabilized methods coupled with finite element methods for a class of parabolic semilinear deterministic and stochastic partial differential equations. Numerical experiments including the semilinear stochastic heat equation with space-time white noise confirm the theoretical findings.
Stabilized explicit methods are particularly efficient for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they loose their efficiency when a severe stiffness is induced by very few "fast" degrees of freedom, as the stiff and nonstiff terms are evaluated concurrently. Therefore, inspired by [A. Abdulle, M. J. Grote, and G. Rosilho de Souza, Preprint (2020), arXiv:2006.00744] we introduce a stochastic modified equation whose stiffness depends solely on the "slow" terms. By integrating this modified equation with a stabilized expli…
This paper discusses a Python interface for the recently published DUNE-FEM-DG module which provides highly efficient implementations of the Discontinuous Galerkin (DG) method for solving a wide range of non linear partial differential equations (PDE). Although the C++ interfaces of DUNE-FEM-DG are highly flexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work easier user interfaces based on Python and the Unified Form Language are provided to open DUNE-FEM-DG for a broader audience. The Python interfaces are demonstrated for both para…
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…
We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form $\m{A}x=λ\m{B}x$, where the matrices $\m{A}$ and/or $\m{B}$ may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilized formulations are used for the numerical approximation of partial differential equations. With the help of classical numerical examples we show that the presence of one (or both) parameters can produce unexpected results.
Second order hierarchical partial least squares regression-polynomial chaos expansion for global sensitivity and reliability analyses of high-dimensional models
Most problems in electrodynamics do not have an analytical solution so much effort has been put in the development of numerical schemes, such as the finite-difference method, volume element methods, boundary element methods, and related methods based on boundary integral equations. In this paper we introduce a local integral boundary domain method with a stable calculation based on Radial Basis Functions (RBF) approximations, in the context of wave chaos in acoustics and dielectric microresonator problems. RBFs have been gaining popularity recently for solving partial differential equations nu…
