A posteriori error analysis of the virtual element method for second-order quasilinear elliptic PDEs

arXiv 2409.17779

In this paper we develop a $C^0$-conforming virtual element method (VEM) for a class of second-order quasilinear elliptic PDEs in two dimensions. We present a posteriori error analysis for this problem and derive a residual based error estimator. The estimator is fully computable and we prove upper and lower bounds of the error estimator which are explicit in the local mesh size. We use the estimator to drive an adaptive mesh refinement algorithm. A handful of numerical test problems are carried out to study the performance of the proposed error indicator.

A framework for implementing general virtual element spaces

SIAM Journal on Scientific Computing 46, no. 3 (2024): B229-B253.

In this paper we present a framework for the construction and implementation of general virtual element spaces based on projections built from constrained least squares problems. Building on the triples used for finite element spaces, we introduce the concept of a virtual element method (VEM) tuple which encodes the necessary building blocks to construct these projections. Using this approach, a wide range of virtual element spaces can be defined. We discuss $H^k$-conforming spaces for $k=1,2$ as well as divergence and curl free spaces. This general framework has the advantage of being easily integrated into any existing finite element package, and we demonstrate this within the open source software package Dune.

Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

IMA Journal of Numerical Analysis 42, no. 2 (2022): 1364-1399.

We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

A higher order nonconforming virtual element method for the Cahn-Hilliard equation

arXiv 2111.11408

In this paper we develop a fully nonconforming virtual element method (VEM) of arbitrary approximation order for the two dimensional Cahn-Hilliard equation. We carry out the error analysis for the semidiscrete (continuous-in-time) scheme and verify the theoretical convergence result via numerical experiments. We present a fully discrete scheme which uses a convex splitting Runge-Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization.

A posteriori error analysis of the virtual element method for quasilinear elliptic PDEs

Computational Methods in Applied Mathematics (CMAM-10) in the minisymposium MS06: Recent advances for adaptivity in applications
University of Bonn, Germany (June 2024)

Virtual element methods for fourth-order problems

Seminar of Numerical Mathematics
Charles University, Czech Republic (February 2023)

Virtual element methods for fourth-order problems

Polytopal Element Methods in Mathematics and Engineering (POEMS) workshop
Politecnico di Milano, Italy (December 2022)

A framework for implementing general higher order virtual element spaces

Computational Methods in Applied Mathematics (CMAM-9)
TU Wien, Austria (August 2022)

Nonconforming virtual element methods for fourth-order problems

8th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS)
Oslo, Norway (June 2022)

A higher order virtual element method for the Cahn-Hilliard equation

Postgraduate Seminar
University of Warwick (May 2022)

A non-virtual exploration of the virtual element method

Celebrating Women in Mathematical Sciences
University of Warwick (May 2022)

Implementing the virtual element method within the DUNE framework

Postgraduate Seminar
University of Warwick, online (June 2021)

Construction of projectors for the conforming and nonconforming virtual element method

British Applied Mathematics Colloquium (BAMC)
University of Glasgow, online (April 2021)

Nonconforming virtual element methods for fourth-order problems

Interplay of Probability, Statistics and Analysis, Joint CDT Colloquium
University of Cambridge, online (December 2020)

The fully nonconforming virtual element method for fourth-order perturbation problems

Statistics, Probability, Analysis & Applied Mathematics (SPAAM) Seminar
University of Warwick, online (November 2020)