Unique features

Space-Time hp-Adaptivity on Dynamical Meshes

Hermes is the first code to provide automatic hp-adaptivity for nonstationary PDE problems on dynamical meshes that are refined and coarsened simultaneously as dictated by the space-time approximation error. The algorithm, which is a combination of the classical Rothe's method and our novel multi-mesh hp-FEM, is extremely robust and works in the same way for virtually all nonstationary PDE problems. For illustration and technical details see the Movie gallery and Publication sections of this home page, respectively.

Adaptive Multi-Mesh hp-FEM

In multi-physics coupled problems, physical fields exhibit dramatical qualitative as well as quantitative differences. For example, boundary layers which are characteristic for viscous flow are not present in electromagnetics, heat transfer, or elasticity. Conversely, electromagnetic field often contains singularities at sharp corners or edges, which do not appear in other physical fields. Standard methods that use the same mesh for all physical fields are thus inefficient. Hermes allows each solution component to be solved on an individual higher-order finite element mesh equipped with an independent automatic hp-adaptivity algorithm. This is done in a monolithic way (without operator splitting) using our novel multi-mesh hp-FEM technique.

Multiple Types of Higher-Order Finite Elements

In multi-physics coupled problems, various solution components live in generally different spaces of functions. Therefore various solution components can be discretized using generally different types of finite elements. Currently, Hermes provides following types of hierarchic higher-order elements:

* Standard H^1^ conforming elements for second-order elliptic/parabolic problems such as elasticity, heat transfer, or electrostatics.
* H(curl) elements (edge elements) to be used, for example, for the electric field.
* H(div) elements (face elements) for the magnetic field, velocity in incompressible viscous flow, etc. (implementation in progress).
* Taylor-Hood P^{k+1}^/P^k^ elements for some applications in incompressible fluid dynamics.
* Discontinuous L^2^ elements that can be used, for example, for pressure in incompressible viscous flow.

Thus in multiphysics coupled problems, Hermes is able to approximate different physical fields using geometrically and polynomially different meshes consisting of different element types. This is done in a monolithic way (i.e., there is one global stiffness matrix for the discrete problem).

hp-FEM with Arbitrary-Level Hanging Nodes

Hermes features our novel algorithm for arbitrary-level hanging nodes. This means that elements of very different sizes can be adjacent to each other in the mesh. This has two basic advantages:

* Small-scale phenomena can be resolved extremely efficiently since no regularity-enforced (unwanted) refinements are present.
* Automatic adaptivity is simplified drastically, since all element refinements are completely local (splitting of an element does not affect the surrounding elements as in standard adaptivity algorithms).

Hermes2D Further Features

* general unstructured triangular and quadrilateral meshes,
* fully anisotropic "hp"-refinement of quadrilateral elements,
* curvilinear elements with edges defined via NURBS,
* algorithms for linear, non-linear, stationary, and time dependent problems,
* easy-to-use automatic "hp"-adaptivity module,
* multiple element types can be used in one computation,
* basic postprocessing and visualization functions.