  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 large expansion
coefficients. Error estimates for this procedure are refined and the behavior
of the procedure is demonstrated via some simple elliptic model problems.
