## Capturing nonstationary reflected shock-wave

We solve compressible Euler equations in a straight channel. This example has a known exact stationary solution which is piecewise constant. The flow is entering with two different Mach numbers and angles from the left and from the top. The following two movies show the evolution of the density (left) and the time-dependent mesh (right).

## Two-component incompressible viscous flow

Navier-Stokes equations for two-component flow consist of the usual Navier-Stokes equations completed by one additional transport equation for the lavel set function. The two liquids have different densities and viscosities, and they are distinguished by the sign of the level set function. The following videos illustrate a sloshing process where initially, the domain is divided vertically into two equal parts, with water on the left side and (lighter) glycol on the right. After the imaginary barrier between the two liquids is removed, the different densities of the fluids initiate motion. The following videos show the evolution of the velocity (left) and level-set function (right).

## Heat and Moisture Transfer in Massive Concrete Walls of a Nuclear Reactor Vessel

We consider an axisymmetric 3D problem consisting of a system of two coupled time-dependent linear parabolic equations describing a simplified model of the transfer of heat and moisture in concrete. The goal of the computation is to predict with high accuracy the distribution of the temperature and moisture in the concrete walls of the reactor vessel after 30 years of operation. In the following videos, the time is not shown linearly – it scales accordingly to the adaptive time step, which ranges from 15 minutes at the beginning of the computation and reaches 1 year later. The following video shows the evolution of the temperature (left) and moisture (right).

## Flame propagation

The underlying model consists of two nonlinear parabolic equations describing the temperature and concentration. The flame moves through the domain from the left to the right. The narrow part serves as a cooler and its function is to slow down the reaction rate. Reaction rate, also called flame intensity, is defined by the Arrhenius law. The following video shows the reaction rate (top) and the corresponding hp-mesh.

## Navier-Stokes equations with heat transfer

We solve incompressible viscous flow in a channel with heated rectangular obstacle. Unknown are both velocity components, pressure, and temperature. This problem is nonlinear, time-dependent, and coupled one way. We solve it with controlled error both in space and time. Note different finite element meshes for the flow and temperature. The Reynolds number is Re=10^5. We use Taylor P2/P1 elements for the flow and hp-meshes for the temperature. Both meshes evolve dynamically in time. The following video shows the velocity magnitude (top) and temperature (bottom).

## Microwave heating

The microwave model consists of a cavity and a small square waveguide attached to its righ-hand side. The cavity contains a food specimen (load) with temperature-dependent electric permittivity. The waves are generated using a sinusoidal current along the right edge of the waveguide. The electrical field deposits energy into the load whose temperature rises and influences the electrical field. This problem is nonlinear, time-dependent, coupled both ways. The following video shows the electric field magnitude and temperature, along with the corresponding meshes.

## Thermoelasticity

This is an old computation which is not space-time adaptive yet (we should redo it soon). We use the plane-strain formulation. A steel construction is heated on top. The rising temperature causes elastic deformations. This problem is linear, time-dependent, coupled one-way.

## Navier-Stokes equations

We solve incompressible viscous flow in a channel with rectangular obstacle, with different values of the Reynolds number. Unknown are both velocity components and the pressure. This problem is nonlinear and time-dependent. We solve it with controlled error both in space and time. Note that the finite element meshes evolve both in space and time.

## Automatic hp-adaptivity in a waveguide

We consider a square waveguide containing circular load. defined using an electric permittivity different from vacuum. The waves are generated using a sinusoidal current in the left edge of the domain. We solve the time-hermonic Maxwell’s equations using higher-order edge elements. The adaptivity process starts from an extremely coarse mesh consisting of only 4 cubic elements. The videos show first 32 steps of the adaptive process.

## Automatic hp-adaptivity in L-shape domain

This video demonstrates progressive reduction of error during automatic hp-adaptivity. The example is related to singular electric field in a domain with re-entrant corner.