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Time and space adaptivity for wave equation

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Olga Gorynina

The wave propagation phenomena are ubiquitous in acoustics, electromagnetism, elasticity theory and their industrial applications. One can cite for example the detection of underwater objects by sonars (acoustic waves) or the design of telecommunication antennas (electromagnetic waves). The computer simulations of these phenomena are based on the solution of partial differential equations known as hyperbolic equations. These computations are performed most often using the methods of finite elements or finite differences on very fine structured meshes. This simplifies the code development but is not necessarily the most efficient option since the important phenomena may be concentrated in a small zone near the wave front and one could thus content oneself with a coarse approximation outside this zone. A strategy that optimizes the use of computing resources is based on the adaptive meshes that follow the wave front.
The automatic construction of adaptive meshes requires the a posteriori error estimates which are now classical in the case of elliptic and parabolic problems discretized in space by the finite elements. This being said, the case of hyperbolic problems remains mostly open. My talk focuses on the a posteriori error analysis for the linear second-order hyperbolic problem discretized by the second order Newmark scheme in time and the finite element method in space.