A practical phase field method for an elliptic surface PDE

An Accelerated Method for Nonlinear Elliptic PDE

We propose two numerical methods for accelerating the convergence of the standard fixed point method associated with a nonlinear and/or degenerate elliptic partial differential equation. The first method is linearly stable, while the second is provably convergent in the viscosity solution sense. In practice, the methods converge at a nearly linear complexity in terms of the number of iterations...

An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDE with Random Coefficients

Generalized polynomial chaos (gPC) methods have been successfully applied to various stochastic problems in many physical and engineering fields. However, realistic representation of stochastic inputs associated with various sources of uncertainty often leads to high dimensional representations that are computationally prohibitive for classic gPC methods. Additionally in the classic gPC methods...

The PDE surface method in higher dimensions

This paper presents a method to extend PDE surfaces to high dimensional spaces. We review a common existing analytic solution, and show how it can be used straightforwardly to increase the dimension of the space the surface is embedded within. We then further develop a numerical scheme suitable for increasing the number of variables that parametrise the surface, and investigate some of the prop...

Low-order dPG-FEM for an elliptic PDE

This paper introduces a novel lowest-order discontinuous Petrov Galerkin (dPG) finite element method (FEM) for the Poisson model problem. The ultra-weak formulation allows for piecewise constant and affine ansatz functions and for piecewise affine and lowest-order Raviart-Thomas test functions. This lowest-order discretization for the Poisson model problem allows for a direct proof of the discr...

The Shadowing Lemma for Elliptic PDE