Sparse approximate solution of partial differential equations

Authors:
Sadegh Jokar;Volker Mehrmann;Marc E. Pfetsch;Harry Yserentant
Affiliations:
Institut für Mathematik, TU Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany;Institut für Mathematik, TU Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany;Institute for Mathematical Optimization, TU Braunschweig, Pockelsstr. 14, 38106 Braunschweig, Germany;Institut für Mathematik, TU Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany
Venue:
Applied Numerical Mathematics
Year:
2010

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Abstract

A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh are refined that belong to nonzero expansion coefficients. Error estimates for this procedure are refined and the behavior of the procedure is demonstrated via some simple elliptic model problems.