ROBUST DISCRETIZATION OF PDES POSED ON CLOSED HYPERSURFACES

Abstract

Partial differential equations (PDEs) posed on surfaces have a multitude of applications ranging from biological membranes and surfactants to active matter and image processing. This dissertation is concerned with the design and analysis of robust numerical schemes for the discretizations of fundamental surface PDEs, such as the heat equation, the tangent Stokes problem, and the transient tangent Stokes system. The first part of the thesis concerns the discretization of the heat equation posed on a closed hypersurface with an Eulerian finite element method called TraceFEM. We modify the existing scheme in a simple but crucial way and, assuming geometric exactness, derive necessary and sufficient conditions for uniform discrete inf-sup stability of this new method. This strong notion of stability leads to a quasi-best approximation property, convergence to minimally regular solutions, and optimal order-regularity error estimates. Consequently, this approach also restores the optimal condition number of the matrix coming from an additional discretization in time. The second part of the thesis is concerned with reformulating fluid flow problems, i.e., (transient) tangent (Navier--)Stokes problems, posed on closed hypersurfaces in a way that allows circumventing the discrete inf-sup condition. By using elementary but nontrivial geometric identities, we derive a pressure Poisson problem perturbed by a lower-order term involving the Ricci curvature map. We propose to replace the divergence equation with the pressure Poisson equation, yielding a system consisting of the momentum equation for the velocity and a Laplace equation for the pressure. The new system no longer has a saddle-point structure, thereby permitting discretization with arbitrary, independent polynomial degrees for the velocity-pressure pair, and thus circumventing the discrete inf-sup condition. To explore this idea, we first study the PDEs induced by the reformulation. We introduce an abstract framework that allows studying reformulations of the tangent Stokes, transient tangent Stokes, and 2D transient tangent Navier--Stokes problems in a systematic way. In particular, this allows deriving well-posedness of the reformulation from the well-posedness of the original problems (some of which we prove to be the case). For linear systems, we further derive improved regularity results and analyze the adjoint problems. This theory allows carrying out rigorous mathematical analysis of the discretization based on the reformulations of the tangent Stokes and the transient tangent Stokes systems. In particular, we prove uniform stability at the discrete level, derive quasi-best approximation properties, and optimal order-regularity error estimates, all while circumventing the discrete inf-sup condition. The reformulation approach allows obtaining new stability and error estimates for the transient tangent Stokes problem that were previously unknown, even in flat domains. We supplement this discussion with numerical experiments corroborating the theory.