Stable Numerical Solution of an Elliptic PDE Inverse Problem Subject to Incomplete Boundary Conditions

This paper addresses the inverse problem of reconstructing complete steady-state solutions for elliptic partial differential equations when boundary information is incomplete a situation common in electromagnetic, thermal, and geophysical modeling where full Dirichlet or Neumann data are not available. The objective is to develop a numerically stable and reproducible method that reconstructs the interior field and missing boundary trace from partial boundary measurements contaminated by noise. To that end, the inverse task is posed as a Tikhonov-regularized optimization problem in which the unknown Dirichlet trace on the unobserved boundary segment minimizes a least-squares discrepancy between forward-model predictions and boundary observations; the forward PDE is discretized by the finite element method and the gradient of the discrete cost is computed via the adjoint-state method, enabling efficient computation independent of the number of boundary parameters. The optimization uses a nonlinear conjugate-gradient scheme with automated L-curve-based regularization selection. Numerical validation on synthetic test cases including a unit-square Poisson problem and a circular-domain example shows that the proposed approach recovers the interior field and the missing boundary trace with low RMSE and small relative L^2 errors for noise levels up to 5%, whereas unregularized reconstructions suffer severe noise amplification. These results demonstrate that the FEM + adjoint + H¹-Tikhonov methodology provides a practical and stable solution approach for elliptic inverse problems with partial boundary conditions, and that this approach scales to moderate resolution meshes and is potentially extendable for non-smooth targets, nonlinear PDEs, and realistic measurement models.