# Dirichlet Boundary Value Correction using Lagrange Multipliers

## TL;DR

This paper introduces a boundary value correction method using Lagrange multipliers for curved boundary approximation, achieving optimal convergence for polynomial degrees up to 3 and providing error estimates related to boundary approximation accuracy.

## Contribution

It presents a novel boundary correction technique with Lagrange multipliers, extending existing methods to curved boundaries with optimal convergence and explicit error bounds.

## Key findings

- Achieves optimal order convergence for polynomial degree up to 3.
- Establishes a relation to Taylor series expansion methods.
- Provides a priori error estimates with explicit meshsize dependence.

## Abstract

We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to the Taylor series expansion approach used by Bramble, Dupont and Tom\'ee [Math. Comp., 26:869--879, 1972] in the context of Nitsche's method and, in the case of inf-sup stable multiplier methods, prove a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.