Fully discrete DPG methods for the Kirchhoff-Love plate bending model

TL;DR

This paper develops fully discrete DPG methods for the Kirchhoff-Love plate bending model, including new formulations and convergence proofs, applicable to non-convex plates with less regular shear forces, supported by numerical validation.

Contribution

It introduces a well-posed, quasi-optimal DPG discretization for the Kirchhoff-Love model, including the gradient of deflection and Fortin operators for lowest-order schemes.

Findings

Convergence orders align with theoretical predictions.

Applicable to non-convex polygonal plates with low regularity shear forces.

Numerical results confirm the effectiveness of the proposed methods.

Abstract

We extend the analysis and discretization of the Kirchhoff-Love plate bending problem from [T. F\"uhrer, N. Heuer, A.H. Niemi, An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation, arXiv:1805.07835, 2018] in two aspects. First, we present a well-posed formulation and quasi-optimal DPG discretization that includes the gradient of the deflection. Second, we construct Fortin operators that prove the well-posedness and quasi-optimal convergence of lowest-order discrete schemes with approximated test functions for both formulations. Our results apply to the case of non-convex polygonal plates where shear forces can be less than $L_2$-regular. Numerical results illustrate expected convergence orders.