= Solution
{c}
For a conforming triangular <finite element mesh>, take
$$
V_h=\{v_h\in C^0(\overline\Omega):v_h|_K\in\mathcal P_2(K)\text{ for every }K\in\mathcal M_h,\quad v_h|_{\Gamma_D}=0\}.
$$
This is the quadratic <conforming finite element space>. Across a common <edge of a polygon>, the restrictions from neighboring <triangles> are univariate <quadratic polynomials>. Their agreement at the two endpoints and the <midpoint> makes their difference a degree-at-most-two <polynomial> with three distinct <roots of a polynomial>, so it vanishes on the entire common <edge of a polygon>. Thus matching the local <finite element degrees of freedom> produces the required continuity. Similarly, setting the three nodal values on every boundary edge carrying the <Dirichlet boundary condition> to zero makes its entire restriction vanish.
Continuity across element interfaces means the piecewise <gradients> give the global <weak gradient>, with no jump contribution, so $V_h\subset V$. There are finitely many local <finite element degrees of freedom>, hence $V_h$ is finite-dimensional. The <Galerkin method> seeks
$$
\boxed{u_h\in V_h,\qquad (\nabla u_h,\nabla v_h)_0+(u_h,v_h)_0=(f,v_h)_0\quad(v_h\in V_h).}
$$
The <bounded bilinear form> and <coercive bilinear form> estimates from part (b) remain valid on $V_h$. Since a <finite-dimensional subspace is closed>, the <Lax-Milgram theorem> gives exactly one $u_h$. Alternatively, the <stiffness matrix> plus the <mass matrix> is a <positive-definite matrix> in any <basis> of $V_h$. The same existence and uniqueness conclusion holds for any finite-dimensional <vector subspace> of this full space.
Back to article page
