# A conforming DG method for the biharmonic equation on polytopal meshes

## TL;DR

This paper introduces a simple, conforming discontinuous Galerkin method for the biharmonic equation on polytopal meshes, providing optimal error estimates and confirming convergence through numerical experiments.

## Contribution

It presents a novel, easy-to-implement conforming DG method for biharmonic problems with proven optimal error estimates.

## Key findings

- Optimal order error estimates in discrete $H^2$ norm.
- Sub-optimal $L^2$ error estimates for lowest order elements.
- Numerical results confirm theoretical convergence.

## Abstract

A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence.