# Linear and Nonlinear Fractional Elliptic Problems

## TL;DR

This paper reviews recent advances in the analysis and numerical methods for linear fractional elliptic problems, focusing on regularity, boundary behavior, and finite element approximations with error estimates and computational challenges.

## Contribution

It provides a comprehensive survey of recent analytical and numerical techniques for fractional elliptic problems, highlighting error analysis and computational strategies.

## Key findings

- Error estimates for finite element methods on graded meshes
- Analysis of boundary regularity and behavior
- Discussion of computational challenges and numerical experiments

## Abstract

This paper surveys recent analytical and numerical research on linear problems for the integral fractional Laplacian, fractional obstacle problems, and fractional minimal graphs. The emphasis is on the interplay between regularity, including boundary behavior, and approximability by piecewise linear finite element methods. We discuss several error estimates on graded meshes, and computational challenges associated to implementing and solving efficiently the ensuing integral equations, along with numerical experiments.