Primal-dual finite element methods for the direct reconstruction of material coefficients in stationary elliptic problems

Abstract We revisit the direct method for reconstructing coefficients in second-order elliptic partial differential equations. Two model problems are considered: first, the reconstruction of the diffusion coefficient in a scalar elliptic problem, and second, the reconstruction of the shear modulus in the elastography problem. To highlight the versatility of the framework, different notions of stability are exploited in the two situations. In the scalar case, the system is interpreted as a hyperbolic transport equation and an inf-sup condition on the discrete level is leveraged for the analysis of the numerical method. We obtain error estimates on the reconstruction coefficient that are suboptimal by half an order, which is known to be sharp on general meshes. In the vector case, the minimization of the residual in dual norm and a stability result on the continuous problem lead to error estimates that are optimal compared to the approximation. For both problems, the theoretical results are illustrated by some numerical examples.