The principal achivement of the theoretical research in our laboratory is a system of unique codes Hermes1D, Hermes2D and Hermes3D, which are free C++/Python libraries for rapid development of adaptive FEM and hp-FEM solvers for partial differential equations (PDEs). These libraries are available under the GPL license (Version 2, 1991). The main strengths of the codes are the following:

• Adaptive hp-FEM methods
• Adaptivity for time-dependent problems on dynamical hp-meshes
• Monolithic discretization of arbitrary multiphysics problems via a novel multimesh hp-FEM

The following are the main features of the Hermes codes in more detail:

• Mature hp-adaptivity algorithms. Hermes puts a major emphasis on credibility of results, i.e., on error control and automatic adaptivity. Practitioners know well how painful it is to use automatic adaptivity in conjunction with standard lower-order approximations such as linear or quadratic elements – the error decreases somehow during a few initial adaptivity steps, but then it slows down and it does not help to invest more unknowns or CPU time. This is typical for low-order methods. In contrast to this, the exponentially-convergent adaptive hp-FEM and hp-DG do not have this problem – the error drops steadily and fast during adaptivity all the way to the desired accuracy.
• Wide applicability. Hermes is completely PDE-independent. Many FEM codes are designed to solve some narrow class of PDE problems (such as elliptic equations, fluid dynamics, electromagnetics etc.). In contrast to that, Hermes does not employ any technique or algorithm that would only work for some particular class of PDE problems. Automatic adaptivity is guided by a universal computational a-posteriori error estimate that works in the same way for any PDE. Of course, this does not mean that it performs equally well on all PDE – some equations simply are more difficult to solve than others. However, Hermes allows tackling an arbitrary PDE or multiphysics PDE system.
• Arbitrary-level hanging nodes. Hermes has a unique original methodology for handling arbitrary-level hanging nodes. This means that extremely small elements can be adjacent to very large ones. When an element is refined, its neighbors are never split forcefully as in conventional adaptivity algorithms. It is well known that approximations with one-level hanging nodes are more efficient compared to regular meshes. However, the technique of arbitrary-level hanging nodes brings this to a perfection.
• Multimesh hp-FEM. Various physical fields or solution components in multiphysics problems can be approximated on individual meshes, combining quality H¹, H(curl) , H(div), and L² conforming higher-order elements. Due to a unique original methodology, no error is caused by operator splitting, transferring data between different meshes.
• Dynamical meshes for time-dependent problems. In time-dependent problems, different physical fields or solution components can be approximated on individual meshes that evolve in time independently of each other. Due to a unique original methodology, no error is caused by transfering solution data between different meshes and time levels. No such transfer takes place in the multimesh hp-FEM – the discretization of the time-dependent PDE system is monolithic.

Selected papers in impacted journals

• T. Vejchodsky, P. Solin, M. Zitka: Modular hp-FEM System HERMES and Its Application to the Maxwell's Equations, Math. Comput. Simul. 76 (2007), pp. 223–228.
• P. Solin, T. Vejchodsky: Higher-Order Finite Elements Based on Generalized Eigenfunctions of the Laplacian, Int. J. Numer. Methods Engrg 73 (2007), pp. 1374–1394.
• P. Solin, T. Vejchodsky, R. Araiza: Discrete Conservation of Nonnegativity for Elliptic Problems Solved by the hp-FEM, Math. Comput. Simul. 76 (2007), pp. 205–210.
• P. Solin, K. Segeth: Hierarchic Higher-Order Hermite Elements on Hybrid Triangular/Quadrilateral Meshes, Math. Comput. Simul. (76) 2007, pp. 198–204.
• M. Zitka, P. Solin, T. Vejchodsky, F. Avila: Imposing Orthogonality to Hierarchic Higher-Order Finite Elements, Math. Comput. Simul. 76 (2007), pp. 211–217.
• P. Solin, T. Vejchodsky: A Weak Discrete Maximum Principle for hp-FEM, J. Comput. Appl. Math. 209 (2007), pp.  54–65.
• T. Vejchodsky, P. Solin: Discrete Maximum Principle for Higher-Order Finite Elements in 1D, Math. Comput. 76 (2007), pp. 1833–1846.
• P. Solin, J. Avila: Equidistributed Error Mesh for Problems with Exponential Boundary Layers, J. Comput. Appl. Math., Vol 218/1 (2008), pp. 157–166.
• P. Solin, J. Cerveny, I. Dolezel: Arbitrary-Level Hanging Nodes and Automatic Adaptivity in the hp-FEM, Math. Comput. Simul. 77 (2008), pp. 117–132.
• T. Vejchodsky, P. Solin: Static Condensation, Orthogonalization of Bubble Functions, and ILU Preconditioning in the hp-FEM, J. Comput. Appl. Math., Vol 218/1 (2008), pp. 192–200.
• P. Kus, P. Solin, I. Dolezel: Solution of 3D Singular Electrostatics Problems Using Adaptive hp-FEM, COMPEL, Vol. 27, Issue 4 (2008), pp. 939–945.
• P. Solin, J. Cerveny, L. Dubcova, D. Andrs: Monolithic Discretization of Linear Thermoelasticity Problems via Adaptive Multimesh hp-FEM, J. Comput. Appl. Math 234 (2010), pp. 2350–2357.
• L. Dubcova, P. Solin, J. Cerveny, P. Kus: Space and Time Adaptive Two-Mesh hp-FEM for Transient Microwave Heating Problems, Electromagnetics, Vol. 30, Issue 1 (2010), pp. 23–40.
• P. Solin, D. Andrs, J. Cerveny, M. Simko: PDE-Independent Adaptive hp-FEM Based on Hierarchic Extension of Finite Element Spaces, J. Comput. Appl. Math. 233 (2010), pp. 3086–3094.
• P. Solin, L. Dubcova, J. Kruis: Adaptive hp-FEM with Dynamical Meshes for Transient Heat and Moisture Transfer Problems, J. Comput. Appl. Math. 233 (2010), pp. 3103–3112.