A two-dimensional Eulerian spline function finite element method for modeling the coupled thermomechanical processes which control lithospheric extension is presented. The model domain comprises a cross section through the lithosphere and part of the asthenosphere. Material may enter or leave through the bottom and side boundaries. On the side boundary a fixed velocity or a fixed force condition can be applied. Highly nonlinear and thermally activated creep laws are used for the brittle and ductile rheologies of the lithosphere. Compositional differences between crust and mantle are described by a continuous function which is advected and which affects density, rheology, and heat production. An approximate topography is calculated from the normal stress at a free-slip surface boundary. A set of sample case studies is presented which highlights the capabilities and possible applications of the method. It also serves to study the influence of numerical parameters and of conceptual assumptions in various nontrivial applications. The examples comprise narrow rifting continuing into the stage of seafloor spreading, formation of a wide basin, formation of a metamorphic core complex, and impact of a mantle plume onto the bottom of the lithosphere. In the last case the entire upper mantle is modeled, and the instability of a thermal bottom boundary layer gives rise to a diapiric mantle plume. The main weakness of the Eulerian approach as compared to the Lagrangian technique was found to be the effective blurring of material discontinuities like the Moho. This is of little consequence in pure shear deformation, but when strong simple shear occurs, it impairs the resolution. The main advantage of the Eulerian technique is that very large deformation poses no problem as it does for the Lagrangian method. The new approach is especially suited to study processes where strong differential flow occurs in part of the modeled system. A further possible advantage of the method is its efficiency in terms of computer time. ¿ American Geophysical Union 1992