**1. Theory of mesh methods for boundary-value problems of mathematical physics**

Methods for constructing and studying mesh schemes for problems with non-smooth solutions; mesh methods of domain decomposition; mixed finite element methods, mesh methods for solving nonlinear degenerate elliptic and parabolic equations; mesh methods for solving nonlinear spectral problems; multigrid methods.

**2. Mathematical models of mechanics and physics**

Methods for studying equations and inequalities arising in the nonlinear theory of liquid and gas filtration, the nonlinear theory of thin elastic shells, the theory of biological membranes, plasma physics, and hydrodynamic lubrication theory; methods for constructing self-similar solutions; numerical methods; methods for solving inverse problems.

**3. Methods for solving variational inequalities**

Construction of mesh approximations of stationary and evolutionary variational inequalities; accuracy estimation; construction and study of efficient iterative methods; techniques for improved approximation of a free boundary; domain decomposition methods for variational inequalities.

**4. Numerical methods in plasma physics**

Numerical models and methods in low-temperature plasma physics: kinetic, hydrodynamic, and hybrid models. Models of interaction between plasma and surfaces: plasma spraying, activation, plasma etching.

**5. Numerical methods in filtration theory**

Numerical methods for solving seepage problems in porous media. Numerical solution of filtration consolidation problems. Numerical methods of multi-phase filtration. Numerical simulation of enhanced oil recovery processes. Numerical methods for solving problems of rational and environmentally friendly mining. Numerical simulation of filtration processes of heavy oils.

**6. Computer simulation and training systems**

Information and communication technologies; computational experiment; numerical modeling of technological processes using modern supercomputers; principles of training systems; simulation of random processes.

**7. Numerical simulation of dynamic processes in multiphase media**

Development of models and methods for calculating dynamic processes in multiphase environments and their applications. Development of models and methods for calculating the dynamics of single gas-vapor bubbles, bubble dynamics in clusters and their applications. Development of models and methods for calculating the pulsed effect of cavitation on the surface of solids and their application.