The field of numerical methods for complex materials and systems is moving towards the development of more efficient and accurate techniques for handling heterogeneous materials, fluid-structure interaction problems, and complex geometries. Partial integration-based regularization and fictitious domain methods are being explored to mitigate the challenges associated with dense system matrices and singular integral kernels. Noteworthy papers in this area include:

• Phantom Domain Finite Element Method, which offers significant advantages in handling complex inclusion geometries and improving computational efficiency.
• Advances on the finite element discretization of fluid-structure interaction problems, which presents a posteriori error estimates for elliptic interface problems.
• On the stability and conditioning of a fictitious domain formulation for fluid-structure interaction problems, which analyzes the behavior of the condition number with respect to mesh refinement.