A quasi-optimal space-time FEM with local mesh refinements for parabolic problems

TL;DR

Addresses the heat equation by developing a quasi-optimal space-time finite element method on locally refined non-tensor meshes. It combines a fractional norms variational framework, a minimal residual discretization, a Scott–Zhang interpolation and Fortin operator to ensure stability, and a multi-level preconditioner with additive subspace correction to enable efficient PCG solves. A posteriori error control is built into the method, and extensive numerical experiments demonstrate improved accuracy and efficiency for smooth, singular, and rough data, including adaptive refinement near domain singularities. The work advances space-time FEM by achieving quasi-optimality on non-trozen meshes and providing practical, scalable solvers for parabolic problems with challenging initial data.

Abstract

We present a space-time finite element method for the heat equation that computes quasi-optimal approximations with respect to natural norms while incorporating local mesh refinements in space-time. The discretized problem is solved with a conjugate gradient method with a (nearly) optimal preconditioner.

Figures (7)

• Figure 1: A partition of a time-space cylinder with $d=1$, where the areas $q_K^t$, $q_{K'}^t$, and $q_{K"}^t$ for time-space cells $K,K',K"$ are marked gray. Notice that Assumption \ref{['ass:Partition']}\ref{['itm:TimeSpaceCylNeighbors']} does not permit the area $q_{K"}^t$, since the left neighbors of $K"$ do not have the same time interval.
• Figure 2: Convergence of the residual $\eta(\mathcal{T})$ and the error $\lVert \nabla_x (u-u_h)\rVert_{L^2(Q)}$ plotted against $\textup{ndof} \coloneqq \dim U_h$ with uniform (solid line) and adaptive (dotted line) mesh refinements with polynomial degree $p_x = 1$ (left) and $p_x = 3$ (right) in the experiment in Section \ref{['subsec:ExpSmootSol']} (Smooth solution). The dashed line (left) indicates the slope $\textup{ndof}^{-3/4}$ and the dash-dotted line (right) the slope $\textup{ndof}^{-1/4}$.
• Figure 3: Meshes at $t = 0$ (left), $t=0.5$ (center), and $t=1$ (right) at level $\ell = 17$ ($\textup{ndof} = 752\, 258$) of the adaptive mesh refinement routine with $p_x = 3$.
• Figure 4: Convergence history plot of the residual $\eta(\mathcal{T})$ in the experiment of Section \ref{['subsec:ExpLShape']} (L-shaped domain) plotted against $\textup{ndof} \coloneqq \dim U_h$ with uniform (solid line) and adaptive (dotted line) mesh refinements. The dash-dotted line indicates the slop $\textup{ndof}^{-1/5}$ and the dashed line indicates the slope $\textup{ndof}^{-1/3}$.
• Figure 5: Left. Convergence history of the residual $\eta(\mathcal{T})$ plotted against $\textup{ndof} = \dim U_h$ for the experiment in Section \ref{['subsec:ExpRoughInit']} (Rough initial data) with uniform (solid) and adaptive (dotted) refinements. The dashed line indicates the rate $\textup{ndof}^{-2/5}$, the dash-dotted line $\textup{ndof}^{-1/8}$, and the dotted line $\textup{ndof}^{-1/4}$. Right. The surface of the mesh resulting from the experiment with $p_x = 3$ and $\textup{ndof} = 13\, 688$.

Theorems & Definitions (55)

• remark 1: Gå rding inequality
• theorem 1: Well-posedness of classical setting
• proposition 1: Equivalent characterizations
• proof
• remark 2: Extension by zero
• theorem 2: Well-posedness of fractional setting
• proof
• lemma 1: Embedding
• proof : Proof of \ref{['itm:embed']}
• theorem 3: Well-posedness with general initial data