Conforming elements and boundary conditions

For a fitted conforming simplicial mesh, continuous piecewise affine functions lie in H¹ because their elementwise weak derivatives have no distributional jump contributions across interior faces. With scalar Lagrange elements of higher degree, matching the full polynomial face traces follows from matching the appropriate shared degrees of freedom. For piecewise smooth functions the essential condition is matching traces almost everywhere on each interior face, not merely checking agreement at one arbitrary point or relying on a visually continuous plot.

A zero-Dirichlet conforming subspace is obtained by imposing zero boundary trace. Other function spaces impose different continuity: H(div) elements match normal components and H(curl) elements match tangential components under their respective transformations. They need not be globally continuous as vector fields. Conformity always refers to a chosen continuous space. A collection of discontinuous cell polynomials is not automatically H¹-conforming just because it is finite-dimensional or has many local degrees of freedom.

For a bounded Lipschitz domain and admissible boundary data g in H to the one-half trace space, choose a lifting G in H¹ with trace g. Write u=G+w, where w has zero trace. The correction satisfies the homogeneous-space variational equation with load F(v)−a(G,v). This is a change of unknown, not a removal of the boundary data. The resulting right-hand side includes the influence of the lifting even when it does not look like an ordinary volume source.

A practical finite element construction uses a discrete lifting G_h whose trace equals the represented boundary data g_h. If that trace approximates rather than equals g, the discrete solution does not live in the exact affine trial set, and a lifting or boundary-data error must enter the analysis. Nodal boundary values require enough regularity to define them; rough trace data may need a suitable projection and stable lifting. One cannot apply homogeneous Céa directly to an error with a nonzero boundary trace and pretend no extra term exists.

Partition an assembled stiffness matrix into interior and prescribed boundary coefficients. The interior system is K_II u_I equal to f_I minus K_IB g_B. Eliminating the known variables in this way preserves the symmetric interior matrix when the variational form is symmetric. Replacing only boundary rows by identity while leaving their columns untouched can destroy symmetry, and zeroing those columns without the matching right-hand-side correction changes the interior equations. The boundary implementation must match the chosen algebraic convention.

Natural Neumann data enter the weak load through the boundary integral after integration by parts, rather than by prescribing every associated value degree of freedom. With only Neumann conditions, constants remain an energy kernel and a source-flux compatibility condition and normalization are needed. Mixed boundary problems require a sufficient essential boundary portion or another mechanism for coercivity. The name of the boundary condition therefore determines both the space and the equations; it is not a cosmetic flag added after assembly.