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This paper seeks to describe the solution of a simple electrostatic problem using an adaptive hp‐FEM and to show the benefits of this approach. Numerical experiments are presented to demonstrate its superiority.

Adaptive hp‐FEM is used. In contrast with standard FEM, the automatic adaptivity procedure can choose from a variety of refinement candidates. An element with over estimated error can be refined in space, or its polynomial degree can be increased. Arbitrary level hanging nodes are allowed, so that no unnecessary refinements are performed in order to keep a mesh regular.

Numerical solution of a singular electrostatic problem is presented. From the comparison it can be seen that the hp‐FEM outperforms both the standard linear and quadratic elements significantly. The accuracy of an hp‐FEM solution would be hard to attain by standard means due to the limited capacity of the computer memory.

The paper describes results obtained from an original and innovative implementation of the adaptive hp‐FEM.

A novel technique for control of complex physical processes based on the solution of their sufficiently accurate models is presented. The technique works with the model order reduction (MOR), which significantly accelerates the solution at a still acceptable uncertainty. Its advantages are illustrated with an example of induction brazing.

The complete mathematical model of the above heat treatment process is presented. Considering all relevant nonlinearities, the numerical model is reduced using the orthogonal decomposition and solved by the finite element method (FEM). It is cheap compared with classical FEM.

The proposed technique is applicable in a wide variety of linear and weakly nonlinear problems and exhibits a good degree of robustness and reliability.

The quality of obtained results strongly depends on the temperature dependencies of material properties and degree of nonlinearities involved. In case of multiphysics problems characterized by low nonlinearities, the results of solved problems differ only negligibly from those solved on the full model, but the computation time is lower by two and more orders. Yet, however, application of the technique in problems with stronger nonlinearities was not fully evaluated.

The presented model and methodology of its solution may represent a basis for design of complex technologies connected with induction-based heat treatment of metal materials.

Proposal of a sophisticated methodology for solution of complex multiphysics problems established the MOR technology that significantly accelerates their solution at still acceptable errors.

The purpose of this paper is to compare different reduced-order models for models of control of induction brazing process. In the presented application, the problem is to reconstruct temperature at the points of interests (hot spots) from information measured at accessible places.

The paper describes the process of induction brazing. It presents the full field model and evaluates the possibilities for obtaining reduced models for temperature estimation. The primary attention is paid to the model based on proper orthogonal decomposition (POD).

The paper shows that for the given application, it is possible to find low-order estimator. In the examined linear case, the best estimator was created using POD reduced model together with the linear Kalman filter.

The authors are aware of two main limitations of the presented study: material properties are considered linear, which is not a completely realistic assumption. However, if strong coupling and nonlinear material parameters are considered, the model becomes unsolvable. The process and measurement uncertainties are not considered.

The paper deals with POD of multi-physics 3 D application of induction brazing. The paper compares 11 different methods for temperature estimator design.

The paper presents the principal elements of automatic adaptivity built in our 2D software for monolithic solution of multiphysics problems based on a fully adaptive finite element method of higher order of accuracy. The adaptive techniques are illustrated by appropriate examples.

Presented are algorithms for realization of the h‐adaptivity, p‐adaptivity, hp‐adaptivity, creation of curvilinear elements for modelling general boundaries and interfaces. Indicated also is the possibility of combining triangular and quadrilateral elements (both classical and curved).

The presented higher‐order adaptive processes are reliable, robust and lead to a substantial reduction of the degrees of freedom in comparison with the techniques used in low‐order finite element methods. They allow solving examples that are by classical approaches either unsolvable or solvable at a cost of high memory and time of computation.

The adaptive processes described in the paper are still limited to 2D computations. Their computer implementation is highly nontrivial (every physical field in a multiphysics task is generally solved on a different mesh satisfying its specific features) and in 3D the number of possible adaptive steps is many times higher.

The described adaptive techniques may represent a powerful tool for the monolithic solution of complex multiphysics problems.

The presented higher‐order adaptive approach of solution is shown to provide better results than the schemes implemented in professional codes based on low‐order finite element methods. Obtaining the results, moreover, requires less time and computer memory.

The purpose of this paper is to present a methodology of high‐precision finite element modeling of induction heating of rotating nonferromagnetic cylindrical billets in static magnetic field produced by appropriately arranged permanent magnets.

The mathematical model consisting of two partial differential equations describing the distribution of the magnetic and temperature fields are solved by a fully adaptive higher‐order finite element method in the monolithic formulation and selected results are validated experimentally.

The method of solution realized by own code is very fast, robust and exhibits much more powerful features when compared with classical low‐order numerical methods implemented in existing commercial codes.

For sufficiently long arrangements the method provides good results even for 2D model. The principal limitation consists in problems with determining correct boundary conditions for the temperature field (generalized coefficient of convective heat transfer as a function of the temperature and revolutions).

The methodology can successfully be used for design of devices for induction heating of cylindrical nonmagnetic bodies by rotation and determination of their operation parameters.

The paper is a presentation of the fully adaptive higher‐order finite element and its utilization for a monolithic numerical solution of a relatively complicated coupled problem.