Related papers: $\Psi$ec: A Local Spectral Exterior Calculus
This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and…
Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite…
We introduce a new class of numerical schemes which allow for low regularity approximations to the expectation $ \mathbb{E}(|u_{k}(\tau, v^{\eta})|^2)$, where $u_k$ denotes the $k$-th Fourier coefficient of the solution $u$ of the…
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…
We present a Parseval tight wavelet frame for the representation and analysis of velocity vector fields of incompressible fluids. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the…
We develop a combinatorial theory of vector bundles with connection on locally ordered simplicial complexes. This is a first step towards a discrete exterior calculus for bundle-valued forms. The basic building block is the discrete…
Constrained by the limitations of learning toolkits engineered for other applications, such as those in image processing, many mesh-based learning algorithms employ data flows that would be atypical from the perspective of conventional…
A new approach is presented to compute the seismic normal modes of a fully heterogeneous, rotating planet. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core. The relevant…
We introduce a novel mesh-free and direct method for computing the shape derivative in PDE-constrained shape optimization problems. Our approach is based on a probabilistic representation of the shape derivative and is applicable for…
This thesis presents the development of a novel finite element library in Rust based on the principles of Finite Element Exterior Calculus (FEEC). The library solves partial differential equations formulated using differential forms on…
In this paper we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vall\'{e}e Poussin type wavelets are able to detect singularities along boundary curves of periodic characteristic functions.…
We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our…
In this article, we propose a new numerical approach to high-dimensional partial differential equations (PDEs) arising in the valuation of exotic derivative securities. The proposed method is extended from Reisinger and Wittum (2007) and…
We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which…
We introduce suitable coordinate systems for pipes and their variants that allow us to transform partial differential equations (PDEs) on the pipe surfaces or in the solid pipes into computational domains with fixed limits/ranges. Such a…
Finite element exterior calculus (FEEC) has been developed as a systematical framework for constructing and analyzing stable and accurate numerical method for partial differential equations by employing differential complexes. This paper is…
The aim of this work is to use systematically the symmetries of the (one dimensional) bacward heat equation with potentiel in order to solve certain one dimensional It\^o's stochastic differential equations. The special form of the drift…
This paper is a continuation of [2], where we complete our partial proof of the Deser-Schwimmer conjecture on the structure of ``global conformal invariants''. Our theorem deals with such invariants P(g^n) that locally depend only on the…
In this paper we extend the novel approach to discrete Painlev\'e equations initiated in our previous work [2]. A classification scheme for discrete Painlev\'e equations proposed by Sakai interprets them as birational isomorphisms between…
Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes…