Related papers: The Adjoint Petrov-Galerkin Method for Non-Linear …
We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may…
A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in \cite{wwnondiv}, the system of equations…
Data-driven reduced-order models often fail to make accurate forecasts of high-dimensional nonlinear dynamical systems that are sensitive along coordinates with low-variance because such coordinates are often truncated, e.g., by proper…
We present a Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e.,…
Stochastic Galerkin methods offer unexplored potential for the numerical simulation of parabolic problems with random variables, in particular if they are combined with variational discretizations of the space and time variables. Due to the…
To describe non-equilibrium transport processes in a quantum device with infinite baths, we propose to formulate the problems as a reduced-order problem. Starting with the Liouville-von Neumann equation for the density-matrix, the…
We present a new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in curved spacetimes. In this paper we assume the background spacetime to be given and static,…
The objective of this paper is to investigate how noisy and incomplete observations can be integrated in the process of building a reduced-order model. This problematic arises in many scientific domains where there exists a need for…
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…
We explore order reduction techniques for solving the algebraic Riccati equation (ARE), and investigating the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a surrogate low dimensional…
We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension…
The application of the Petrov-Galerkin projection method in Cosserat rod finite element formulations offers significant advantages in simplifying the expressions within the discrete virtual work functionals. Moreover, it enables a…
Projection-based model reduction is among the most widely adopted methods for constructing parametric Reduced-Order Models (ROM). Utilizing the snapshot data from solving full-order governing equations, the Proper Orthogonal Decomposition…
In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of…
In this paper we present a new multiscale discontinuous Petrov--Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of Petrov--Galerkin version of…
This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework…
We present a method for linear stability analysis of systems with parametric uncertainty formulated in the stochastic Galerkin framework. Specifically, we assume that for a model partial differential equation, the parameter is given in the…
Galerkin and Petrov-Galerkin methods are some of the most successful solution procedures in numerical analysis. Their popularity is mainly due to the optimality properties of their approximate solution. We show that these features carry…
This paper presents an auto-stabilized weak Galerkin (WG) finite element method for the Biot's consolidation model within the classical displacement-pressure two-field formulation. Unlike traditional WG approaches, the proposed scheme…
In the construction of reduced-order models for dynamical systems, linear projection methods, such as proper orthogonal decompositions, are commonly employed. However, for many dynamical systems, the lower dimensional representation of the…