Exact local finite Riesz solves for practical DPG tests

Inputs: A finite-dimensional real graph-test space presented as $V_r(K)=\mathrm{span}\{v_1,\dots ,v_N\}$, an ordered finite generating family (not assumed independent), a trial datum $w$, the graph inner product $(\cdot ,\cdot {)}_V$, and the linear functional ${\mathrm{\ell }}_w(v)=b_K(w,v)$ on $V_r(K)$.

Output: Either a vector in $V_r(K)$ or the status singular-test-matrix.

Admissible conditions: $N\ge 0$; all entries are real and exact; ${\mathrm{\ell }}_w$ is defined and linear on $V_r(K)$. The Gram entries and functional values are evaluated exactly. No quadrature or rounded solve is part of this procedure.

Instructions:

1. Form $G_{ij}=(v_j,v_i{)}_V$ and $g_i={\mathrm{\ell }}_w(v_i)$ exactly.
2. Initialize $L=I_N$ and compute an unpivoted symmetric $LD{L}^{\mathsf{T}}$ factorization in the listed order. At pivot $k$, set $d_k=G_{kk}-{\sum }_{j<k}{L}_{kj}^2{d}_j\mathrm{.}$ If $d_k\le 0$, return singular-test-matrix. Otherwise set $D_{kk}=d_k$ and $L_{ik}=\frac{G_{ik}-{\sum }_{j<k}{L}_{ij}{L}_{kj}{d}_j}{d_k}(i>k)\mathrm{.}$
3. Solve $Lz=g$, $Dy=z$, and $L^{\mathsf{T}}c=y$ exactly, and return ${\sum }_{j=1}^N{c}_j{v}_j$. For $N=0$, return the zero vector of $V_r(K)$.