Among the current (at the end of 2019) trends regarding the further development of the finite element method, the following can be distinguished:

1. Scientific research on the development of better and better methods of stabilizing difficult simulations for which the classical Galerkin method behaves unstably. Methods based on minimizing the residual are particularly strongly developed, in which stabilization is achieved by defining the testing space larger than the approximation space. In the adaptive finite element method, the DPG stabilization method is the most popular. The recently discovered Fortin operator gives hope for precise determination of which testing space should be selected for a given approximation space in order to obtain automatic stabilization of the solution. Stabilization methods based on the discontinuous Galerkin formulation are also developing particularly intensively.
2. Deep neural networks are also of interest to the world of simulation using the finite element method. Research is being carried out on the use of neural networks to automatically find optimal testing functions in the Petrov-Galerkin formulation by the method of minimizing the residual in order to obtain automatic stabilization of the simulation. Scientists are also interested in the use of neural networks for automatic decision-making on the adaptation of computational grids.
3. The isogeometric finite element method is also enjoying growing popularity. This is due to the fact that in Computer Aided Design (CAD) systems, the design of objects on which simulations are carried out using the finite element method is performed using functions derived from the B-spline family, such as NURBS (Non-Uniform Rational B- splines), enabling the faithful reproduction of the circle geometry, T-splines, hierarchical B-spline functions, TH B-splines and LR B-splines. All these families try to solve problems with B-spline functions such as the ability to span on adaptive meshes, the ability to mix splines of different order, and perform local hp adaptations so that the "partition of unity" property of the splines is preserved (meaning the summing of B-spline basis functions to one at each point in the domain). The integration of computer simulations with design allows you to bypass the costly stage of transforming the object model described with the use of functions from the spline family into a computational mesh covered with finite elements, often tetrahedral, and not having much in common with the geometry described in CAD systems. The authors of isogeometric analysis protect many aspects related to the integration of CAD systems with Computer Aided Engineering (CAE) systems by patent rights.
4. Parallel simulations are carried out on a large scale using the finite element method for non-stationary, often non-linear problems. The most frequently used algorithms of alternating-direction solvers and iterative solvers based on multi-grid methods, which combine projection elements between grids of different sizes, and iterative solvers. Increasing the computing power of computers enables the use of the finite element method in conjunction with inverse algorithms. In particular, model parameters are sought, such as the optimal position of pumps in the liquid fuel fossil extraction problem, in order to maximize extraction and minimize environmental pollution (e.g., groundwater).
5. The finite element method has found application in many areas of life. It is used for the design and analysis of cars, airplanes, mobile phones and building structures. It is used in medical simulations such as modeling blood flow in the central circulatory system, modeling tumor growth and modeling the propagation of electromagnetic waves to identify lesions. It is used in simulations of weather, climate, pollution propagation, and in simulations of extreme phenomena such as typhoons or earthquakes.

The finite element method is a simulation tool. Like many tools, it can be used in many different ways. It is a powerful tool that can model reality and change the world we live in. We should always remember about the ethical messages resulting from our choices and goals we pursue, especially when we have such a powerful tool in our hands.