# Adaptive solver: error and shutoff criterions

## Introduction

Within the framework of an adaptive solver, it is necessary to use:

• An error criterion to detect the meshing areas which are too loose
• A stop signal criterion to set a term to the iterative process in the adaptive solver

## Error criterion

In electrostatic, the error criterion is based on the Maxwell-Gauss equation:

In magnetostatics, the error criterion is based on the Maxwell-Ampère equation:

In Steady state AC magnetic, for the conducting region, the error criterion is based on pointing vector. For non conducting region, like in magnetostatics, this is the Maxwell Ampere equation , which is preserved.

From a numerical viewpoint, it should be verified that the accuracy of the equations « weakly » resolved should be satisfactory.

## Error criterion in 3D

Adaptive solver detects the locations to refine through an error map. The error criterion depends on the application:

• In Electro Static 3D, the error criterion is based on the Maxwell-Gauss equation:

• In Magneto Static 3D, the error criterion is based on the conservation of the magnetic flux density:

This error criterion ensures the precision of the equation “weakly” solved.

## Shutoff criterion

Currently, there are two stop signal criteria:

• The error threshold based on the energy. This permits the study of the evolution of the energy over a region by calculating the relative energy error from an iteration to another
• The maximum number of iterations of the adaptive solver

## How to choose the threshold?

The value of the error threshold is defined either by the user, or automatically by Flux:

• A weak threshold (s = 0.25) is equivalent to a high accuracy
• An average threshold (s = 0.50) is equivalent to an average accuracy
• A strong threshold (s = 0.75) is equivalent to a weak accuracy