**WavES** (Wave Equations Solutions) is a combined theoretical and practical tool for the numerical solution of different types of time-dependent Wave Equations (acoustic, elastic and electromagnetic). The theoretical part consists of published books, papers, courses and presentations, where new efficient numerical methods and strategies for the solution of time-dependent wave equations are presented. The practical part is represented by the C++ program library WavES for the computational solution of time-dependent wave equations (acoustic, elastic and electromagnetic) using three different methods: Finite Element Method (FEM), Finite Difference Method (FDM), Hybrid FEM/FDM method.

**WavES** started to be developed in 2000-2003 at Finite Element Center in Gothenburg, Sweden, at Chalmers University of Technology and Gothenburg University. In 2003-2005, Dr. Larisa Beilina continued to work with WavES at Basel University (Switzerland) and in 2007-2008 at NTNU-Trondheim (Norway).

She has extended WavES with classes for the solution of the forward problem for the electromagnetic wave equation in 2D and 3D using FDTD, stabilized FEM and hybrid FEM/FDM method. In addition, there were developed a new set of classes for the solution of coefficient inverse problems for all above named equations using the adaptive finite element method (see section Applications). Nowadays WavES Project is hosted at the Department of Mathematical Sciences of Chalmers University of Technology and Gothenburg University. Dr. Vladimir Timonov from SibSUTIS and Novosibirsk State University (Novosibirsk, Russia) joined the project in December 2011.

**WavES** uses PETSc 2 and PETSc 3 libraries which are a suite of data structures and routines for the scalable (parallel) solution of scientific applications modelled by partial differential equations. It employs the MPI standard for parallel implementation.

## Features

The main aim of WavES Project is to enable efficient development of modern finite element/difference codes for solution of forward problems for different types of wave equation (acoustic, elastic and electromagnetic) using structured FDM meshes, adaptively refined FEM meshes and hybrid FEM/FDM meshes. The hybrid FEM/FDM method means that different numerical methods, finite elements and finite differences, are used in different subdomains. The purpose is to combine the flexibility of finite elements with the efficiency of finite differences. The hybrid approach may be an important tool to reduce the execution time and memory requirement for large scale computations.

Computations in WavES are performed on a parallel infrastructure in the center for scientific and technical computing C3SE at Chalmers University of Technology, Gothenburg, Sweden.

## Applications

WavES Project has been successfully used for the solution of Partial Differential Equations (PDE) in various fields of computational mathematics. Some of these applications are:

- Scanning acoustic microscopy to reconstruct elastic properties of the bones.
- Reconstruction of dielectrics on transmitted experimental data in approximate globally convergent algorithm.
- Solution of 3D coefficient inverse problem for the Maxwell’s system in time domain.
- Subsurface imaging.