We consider Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. We consider abstract splitting methods associated with this decomposition where no discretization in space is made. We prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size. This result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time, provided the initial data is small enough. This result holds for numerical schemes controlling the round-off error at each step to avoid possible high frequency energy drift. We apply this results to nonlinear Schrödinger equations as well as the nonlinear wave equation.}
We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equatio…
We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved a…
Numerical Construction of Quasi-Periodic Solutions for Nonlinear PDEs: I. Bounded Perturbation
Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学与交叉科学研究院）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉科学研究院）
Sampling recovery in Bochner spaces and applications to parametric PDEs
Mathematisch-Geographische Fakultät, KU Eichstätt-Ingolstadt（凯泽大学艾希施泰特-因戈尔施塔特数学地理学院）；Information Technology Institute, Vietnam National University, Hanoi（越南国家大学河内信息技术研究所）
Splitting methods for Intermediate Long Wave and perturbed Benjamin--Ono models
Nantes Université（南特大学）；CNRS（法国国家科学研究中心）；Institut de Recherche Mathématique avancée（高等数学研究所）；Université de Strasbourg（斯特拉斯堡大学）；INRIA Nancy-Grand Est（南锡大东部国家信息与自动化研究所）
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). In particular, most of the numerical approximation schemes studied in the scientific literature suffer under the curse of dimensionality in the sense that the number of computational operations needed to compute an approximation with an error of size at most $ \varepsilon > 0 $ grows at least exponentially in the PDE dimension $ d \in \mathbb{N} $ or in the reciprocal of $ \varepsilon $. Recently, so-called full-history recursive multilevel Picard (…
Medusa, a novel library for implementation of strong form mesh-free methods, is described. We identify and present common parts and patterns among many such methods reported in the literature, such as node positioning, stencil selection and stencil weight computation. Many different algorithms exist for each part and the possible combinations offer a plethora of possibilities for improvements of solution procedures that are far from fully understood. As a consequence there are still many unanswered questions in mesh-free community resulting in vivid ongoing research in the field. Medusa implem…
Multigrid methods are popular for solving linear systems derived from discretizing PDEs. Local Fourier Analysis (LFA) is a technique for investigating and tuning multigrid methods. P-multigrid is popular for high-order or spectral finite element methods, especially on unstructured meshes. In this paper, we introduce LFAToolkit.jl, a new Julia package for LFA of high-order finite element methods. LFAToolkit.jl analyzes preconditioning techniques for arbitrary systems of second order PDEs and supports mixed finite element methods. Specifically, we develop LFA of p-multigrid with arbitrary second…
