We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with fractional derivative order $\alpha\in(0,1)$ in time. It relies on three technical tools: version discrete Grönwall type inequality, maximal regularity, and regularity theory equations. establish criterion showing inequality verify it L1 scheme convolution quadrature generated by backward difference formulas. Further, we provide complete solution...

We develop proper correction formulas at the starting $k-1$ steps to restore desired $k$th-order convergence rate of $k$-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The can be achieved even if source term is not compatible with initial data, which allowed nonsmooth. provide complete error estimates subdiffusion case $\alpha\in (0,1)$ and sketch proof diffusion-wave $\alpha\in(1,2)$. Extensive numerical examples are...

We construct and analyze a class of extrapolated linearized Runge--Kutta (RK) methods, which can be arbitrarily high order, for the time discretization Allen--Cahn Cahn--Hilliard phase field equations, based on scalar auxiliary variable (SAV) formulation. prove that proposed $q$-stage RK--SAV methods have $q$th-order convergence in satisfy discrete version energy decay property. Numerical examples are provided to illustrate property accuracy methods.

In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with linearized semi-implicit Euler scheme for equations incompressible miscible flow in porous media. We prove that optimal $L^2$ hold without any time-step (convergence) conditions, while all previous works require certain restrictions. Our theoretical results provide new understanding on commonly used schemes. The proof is based splitting into two parts: from time discretization PDEs finite...

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, $d=2,3$. In our analysis, we split function into two parts, one from spatial discretization temporal discretization, by introducing a corresponding time-discrete (elliptic) system. We present rigorous regularity solution system estimates time discretization. With these proved...

For characterizing the Brownian motion in a bounded domain: $Ω$, it is well-known that boundary conditions of classical diffusion equation just rely on given information solution along domain; contrary, for Lévy flights or tempered domain, involves complementary set i.e., $\mathbb{R}^n\backslash Ω$, with potential reason paths corresponding stochastic process are discontinuous. Guided by probability intuitions and perspectives anomalous diffusion, we show reasonable ways, ensuring clear...

In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with time-dependent diffusion coefficient, obtained by Galerkin finite element method conforming piecewise linear elements in space and backward Euler convolution quadrature time. The regularity model proved both nonsmooth initial data incompatible source term. Optimal-order convergence numerical established using proven solution novel perturbation argument via freezing coefficient...

.A new class of parametric finite element methods, with a type artificial tangential velocity constructed at the continuous level, is proposed for solving surface evolution under geometric flows. The method by coupling normal flow an determined harmonic map from fixed reference \(\mathcal{M}\) to unknown \(\Gamma (t)\), formulated level as system partial differential equations in terms Lagrange multiplier. Since almost angle-preserving, could preserve mesh quality, i.e., shapes triangles,...

.A class of high-order mass- and energy-conserving methods is proposed for the nonlinear Schrödinger equation based on Gauss collocation in time finite element discretization space, by introducing a energy-correction post-process at every level. The existence, uniqueness, convergence numerical solutions are proved. In particular, error solution \(O(\tau^{k+1}+h^p)\) \(L^\infty (0,T;H^1)\) norm after incorporating accumulation errors arising from post-processing correction procedure all...

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

In this work, we establish the maximal [Formula: see text]-regularity for several time stepping schemes a fractional evolution model, which involves derivative of order text], in time. These include convolution quadratures generated by backward Euler method and second-order difference formula, L1 scheme, explicit variant Crank-Nicolson method. The main tools analysis operator-valued Fourier multiplier theorem due to Weis (Math Ann 319:735-758, 2001. doi:10.1007/PL00004457) its discrete...

In this work, we analyse a Crank-Nicolson type time-stepping scheme for the subdiffusion equation, which involves Caputo fractional derivative of order |$\alpha\in (0,1)$| in time. It hybridizes backward Euler convolution quadrature with |$\theta$|-type method, parameter |$\theta$| dependent on |$\alpha$| by |$\theta=\alpha/2$| and naturally generalizes classical Crank–Nicolson method. We develop essential initial corrections at starting two steps and, together Galerkin finite element method...

A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The method consists a $k$th-order multistep exponential integrator in time and lumped mass finite element space piecewise $r$th-order polynomials Gauss--Lobatto quadrature. At every level, extra values violating are eliminated at nodal points by cut-off operation. remaining satisfy proved be convergent an error bound...

A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on scalar auxiliary variable formulation, which consists a Gauss collocation temporal discretization and spatial discretization. The proved to be well-posed conserving both mass energy at level. An error bound form $O(h^p+\tau^{k+1})$ in $L^\infty(0,T;H^1)$-norm is established, where $h$ $\tau$ denote mesh sizes, respectively, $(p,k)$ degree elements....

In this paper, we study linearized Crank--Nicolson Galerkin finite element methods for time-dependent Ginzburg--Landau equations under the Lorentz gauge. We present an optimal error estimate schemes (almost) unconditionally (i.e., when spatial mesh size $h$ and temporal step $\tau$ are smaller than a given constant), while previous analyses were only some with strong restrictions on time step-size. The key to our analysis is boundedness of numerical solution in norm. prove cases $\tau\ge h$...

It is shown that for a parabolic problem with maximal $L^p$-regularity (for $1<p<\infty$), the time discretization by linear multistep method or Runge--Kutta has $\ell^p$-regularity uniformly in stepsize if A-stable (and satisfies minor additional conditions). In particular, implicit Euler method, Crank--Nicolson second-order backward difference formula (BDF), and Radau IIA Gauss methods of all orders preserve regularity. The proof uses Weis' characterization terms $R$-boundedness resolvent,...

We analyze fully implicit and linearly backward difference formula (BDF) methods for quasilinear parabolic equations, without making any assumptions on the growth or decay of coefficient functions. combine maximal regularity energy estimates to derive optimal-order error bounds time-discrete approximation solution its gradient in maximum norm norm.

We analyze fully implicit and linearly backward difference formula (BDF) methods for quasilinear parabolic equations, without making any assumptions on the growth or decay of coefficient functions. combine maximal regularity energy estimates to derive optimal-order error bounds time-discrete approximation solution its gradient in maximum norm norm.

An exponential type of convolution quadrature is proposed as a time-stepping method for the nonlinear subdiffusion equation with bounded measurable initial data. The combines contour integral representation solution, approximation integrals, multistep integrators ordinary differential equations, and locally refined stepsizes to resolve singularity. $k$-step can have $k$th-order convergence solutions based on natural regularity solution