Bend 3d Mixed Virtual Element Method for Elliptic Problems

TL;DR

This paper introduces a virtual element method for 3D elliptic problems that effectively handles curved geometries, ensuring high accuracy and mass conservation without degradation due to geometric approximation errors.

Contribution

It presents a novel virtual element scheme for 3D elliptic problems that naturally manages curved elements and computes polynomials over them, improving accuracy in complex geometries.

Findings

Handles curved geometries without accuracy loss

Provides mass conservative fluxes directly

Demonstrates theoretical and numerical validation

Abstract

In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty, here proposed, is that curved elements are naturally handled without any degradation of the solution accuracy. In fact, in presence of curved boundaries, or internal interfaces, the geometrical error introduced by planar approximations may dominate the convergence rate limiting the benefit of high-order approximations. We consider the Darcy problem in its mixed form to directly obtain, with our numerical scheme, accurate and mass conservative fluxes without any post-processing. An important step to derive this new scheme is the actual computation of polynomials over curved polyhedrons, here presented and discussed. Finally, we show the theoretical analysis of the scheme as well as several numerical examples to support our findings