Maximum-norm stability and maximal L^p regularity of FEMs for parabolic equations with Lipschitz continuous coefficients

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximum-norm stability of the semigroup generated by the corresponding elliptic finite element operator, and prove the space-time stability of the parabolic projection onto the finite element space in $L^\infty(\Omega_T)$ and $L^p((0,T);L^q(\Omega))$, $1<p,q<\infty$. The maximal $L^p$ regularity of the parabolic finite element equation is also established

Convergence of a decoupled mixed FEM for the dynamic Ginzburg--Landau equations in nonsmooth domains with incompatible initial data

In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg--Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform $L^{3+\delta}$ regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the N\'ed\'elec finite element space, and introducing a $\ell^2(W^{1,3+\delta})$ estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge

A singular integral approach to the maximal L^p regularity of parabolic equations

Here we describe a simple and fundamental approach to the maximal L^p regularity of parabolic equations, which only uses the concept of singular integrals of Volterra type. Knowledge of analytic semigroups, R-boundedness or H^\infty-functional calculus are not required

Global well-posedness of the time-dependent Ginzburg-Landau superconductivity model in curved polyhedra

We study the time-dependent Ginzburg--Landau equations in a three-dimensional curved polyhedron (possibly nonconvex). Compared with the previous works, we prove existence and uniqueness of a global weak solution based on weaker regularity of the solution in the presence of edges or corners, where the magnetic potential may not be in $L^2(0,T;H^1(\Omega)^3)$

Maximal Lp\bf L^p analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra

The paper is concerned with Galerkin finite element solutions for parabolic equations in a convex polygon or polyhehron with a diffusion coefficient in $W^{1,N+\epsilon}$ for some $\epsilon>0$, where $N$ denotes the dimension of the domain. We prove the analyticity of the semigroup generated by the discrete elliptic operator, the discrete maximal $L^p$ regularity and the optimal $L^p$ error estimate of the finite element solution for the parabolic equation

Maximum norm analysis of implicit-explicit backward difference formulas for nonlinear parabolic equations

We establish optimal order a priori error estimates for implicit-explicit BDF methods for abstract semilinear parabolic equations with time-dependent operators in a complex Banach space settings, under a sharp condition on the non-self-adjointness of the linear operator. Our approach relies on the discrete maximal parabolic regularity of implicit BDF schemes for autonomous linear parabolic equations, recently established in [20], and on ideas from [7]. We illustrate the applicability of our results to four initial and boundary value problems, namely two for second order, one for fractional order, and one for fourth order, namely the Cahn-Hilliard, parabolic equations

Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations

This paper is concerned with the time-step condition of commonly-used linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin finite element approximations. In particular, we study the time-dependent nonlinear Joule heating equations. We present optimal error estimates of the semi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm without any time-step restriction. Theoretical analysis is based on a new splitting of the error and precise analysis of a corresponding time-discrete system. The method used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations for which previous works often require certain restriction on the time-step size $\tau$

Unconditionally optimal error analysis of fully discrete Galerkin methods for general nonlinear parabolic equations

The paper focuses on unconditionally optimal error analysis of the fully discrete Galerkin finite element methods for a general nonlinear parabolic system in $\R^d$ with $d=2,3$. In terms of a corresponding time-discrete system of PDEs as proposed in \cite{LS1}, we split the error function into two parts, one from the temporal discretization and one the spatial discretization. We prove that the latter is $\tau$-independent and the numerical solution is bounded in the $L^{\infty}$ and $W^{1,\infty}$ norms by the inverse inequalities. With the boundedness of the numerical solution, optimal error estimates can be obtained unconditionally in a routine way. Several numerical examples in two and three dimensional spaces are given to support our theoretical analysis

Uniform BMO estimate of parabolic equations and global well-posedness of the thermistor problem

Global well-posedness of the time-dependent (degenerate) thermistor problem remains open for many years. In this paper, we solve the problem by establishing a uniform-in-time BMO estimate of inhomogeneous parabolic equations. Applying this estimate to the temperature equation, we derive a BMO bound of the temperature uniform with respect to time, which implies that the electric conductivity is a $A_2$ weight. The H\"{o}lder continuity of the electric potential is then proved by applying the De Giorgi--Nash--Moser estimate for degenerate elliptic equations with $A_2$ coefficient. Uniqueness of solution is proved based on the established regularity of the weak solution. Our results also imply the existence of a global classical solution when the initial and boundary data are smooth

Unconditionally optimal error estimates of a Crank--Nicolson Galerkin method for the nonlinear thermistor equations

This paper focuses on unconditionally optimal error analysis of an uncoupled and linearized Crank--Nicolson Galerkin finite element method for the time-dependent nonlinear thermistor equations in $d$-dimensional space, $d=2,3$. We split the error function into two parts, one from the spatial discretization and one from the temporal discretization, by introducing a corresponding time-discrete (elliptic) system. We present a rigorous analysis for the regularity of the solution of the time-discrete system and error estimates of the time discretization. With these estimates and the proved regularity, optimal error estimates of the fully discrete Crank--Nicolson Galerkin method are obtained unconditionally. Numerical results confirm our analysis and show the efficiency of the method