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The Poisson--Boltzmann equation is widely used to model electrostatics inmolecular systems. Available software packages solve it using finitedifference, finite element, and boundary element methods, where the latter isattractive due to the accurate representation of the molecular surface andpartial charges, and exact enforcement of the boundary conditions at infinity.However, the boundary element method is limited to linear equations andpiecewise constant variations of the material properties. In this work, wepresent a scheme that couples finite and boundary elements for thePoisson--Boltzmann equation, where the finite element method is applied in aconfined {\it solute} region, and the boundary element method in the external{\it solvent} region. As a proof-of-concept exercise, we use the simplestmethods available: Johnson--N\'ed\'elec coupling with mass matrix and diagonalpreconditioning, implemented using the Bempp-cl and FEniCSx libraries via theirPython interfaces. We showcase our implementation by computing the polarcomponent of the solvation free energy of a set of molecules using a constantand a Gaussian-varying permittivity. We validate our implementation against thefinite difference code APBS (to 0.5\%), and show scaling from protein G B1 (955atoms) up to immunoglobulin G (20\,148 atoms). For small problems, the coupledmethod was efficient, outperforming a purely boundary integral approach. ForGaussian-varying permittivities, which are beyond the applicability of boundaryelements alone, we were able to run medium to large sized problems on a singleworkstation. Development of better preconditioning techniques and the use ofdistributed memory parallelism for larger systems remains an area for futurework. We hope this work will serve as inspiration for future developments formolecular electrostatics with implicit solvent models.

Coupling finite and boundary element methods to solve the Poisson--Boltzmann equation for electrostatics in molecular solvation