Rajen Kumar Sinha, Professor, Department of Mathematics, Indian Institute of Technology Guwahati, for the award of the degree of Doctor of Philosophy and this work has not been submitted elsewhere for the degree. A special model for standard diffusion with memory, also known as the parabolic integro-differential equation (PIDE), has the form
Existing Well-posedness Results and Numerical Schemes for SEM
The authors have used the high-order orthogonal spline collocation method for the spatial discretization and the L1 approximation together with the ADI method in time to establish an optimal order error estimate. We wish to emphasize that due to the presence of the Volterra integral term and the Caputo fractional derivative, such expansions are not straightforward.
Notations and Preliminaries
In the stability and regularity estimates of SEM (1.5) we will make use of the following important inequality. For simplicity, in the rest of the thesis we will write u(x, t) as u(t) and there will be no confusion.
Contributions and Outline of the Thesis
In this chapter, we prove the existence and uniqueness of the solution to problem (1.5) and establish the a priori bounds of the solution under various correctness assumptions on the initial data and source function. In this regard, we prove some more general stability results for solutions of homogeneous and inhomogeneous problems.
Existence, Uniqueness and Stability Results
The following theorem represents the existence of a unique solution (2.1) and stability properties of the solution for smooth initial data. The uniqueness of the solution (2.1) is a consequence of Lemma 2.2.2 and the standard argument for the existence of a unique solution of the Volterra integral equation.
Infinite Differentiability of the Solution
Further, the inequality (2.23) reveals the smoothing property of the homogeneous SEM with respect to space for positive time when u0 ∈L2(Ω). The following lemma plays an important role in proving the infinite differentiability with respect to the solution time of the homogeneous problem. We are now ready to present the theorem concerning the infinite differentiability of the solution of the homogeneous equation (f = 0 in (2.1)) with respect to time.
General Stability Results
Wealth 34 In the following we prove some stability results for the solution of the non-homogeneous problem. The first theorem presents stability estimates for the Caputo derivative, while the second describes those for the usual partial derivative of the continuous solution. Here the coefficients cij = cji and c0 ≥ 0 are smooth functions of the space variable alone and cij form a uniform positive definite matrix on Ω.
Concluding Remarks
We prove optimal order error bounds in both L2(Ω) and H1(Ω) norms for smooth initial data using both Ritz and Ritz-Volterra projections. Here we study the spatially discrete finite element approximation to problem (3.1) and derive the error estimates for both smooth and non-smooth initial data. In Section 3.2, we develop the semi-discrete GFEM and derive error bounds for both smooth and non-smooth initial data.
Standard Galerkin Finite Element Method
Smooth Data Error Estimates
Using the Ritz projection as the comparison function, we partition the error u(t)−uh(t) as usual as for parabolic problems. Hence, the bounds for θ in both L2(Ω) and H1(Ω) norms are not straightforward from the approximation properties of the Ritz-Volterra projection Vh. With gh = Rhg, the Ritz projector of g, we apply the Cauchy-Schwarz inequality and the gradient norm estimate of ∂αtρ(t) to obtainˆ.
Nonsmooth Data Error Estimates
Then there exists a positive constant C such that . i) The main advantage of using the Ritz-Volterra projection as a comparison function is that the term Rt. 3.18)), which greatly simplifies the analysis.
Optimal Error Estimates for Nonsmooth Data
For further development in the direction of error analysis, it is important to obtain the limit of the function. Since cos ˜ϑ is a constant, it can be seen that the minimum value of the expression is 1 + 2rcos ˜ϑ+r2 sin2ϑ. In the following theorem, we prove error estimates that are independent of the factor |lnh|.
Numerical Illustrations
Concluding Remarks
Since the energy method is the most basic technique in a priori analysis, it is therefore natural to study the a priori error analysis for SEM (1.5) with energy arguments. Moreover, we also impose a near-optimal point-in-time bounded error at the maximum rate for smooth initial data. In Section 4.2, we introduce some notations and preliminaries which are useful in convergence analysis.
Notations and Preliminary Results
Semidiscrete error analysis by energy arguments 63 In addition to the above, we require the following continuity property of Iα [52, Lemma 3.1(iii)]: For g, φ∈L2(0, T;L2(Ω)) and any positive. Semidiscrete error analysis by energy arguments 64 Here u0h is an approximation of u0 in Sh and in the convergence analysis we take u0h =Phu0, wherePh :L2(Ω) →Sh is the standard L2 projection defined by. In addition to the above, for the maximum norm estimate, we need the following discrete Sobolev inequality.
Convergence Analysis
Semidiscrete Error Analysis by Energy Arguments 65 Thus we are left with the task of bounding kϑkan and k∇ϑk for both smooth and non-smooth initial data. Semidiscrete error analysis by having energy arguments 69 the inequality (4.3), and Lemmas in the above inequality. Semi-discrete error analysis by energy arguments 70 Repeating the steps of the term T1 as in Lemma 4.3.1, we write.
Numerical Illustrations
To calculate the CRs of the semidiscrete solution, we use the Euler convolution quadrature scheme backward to discretize the fractional derivative and the (left) orthogonal rule for the integral term. In order to calculate the errors for the L2(Ω)- and H1(Ω)-norms and the associated CRs, we calculate the discrete solution (we discretize the partial derivative and the integral. Since the non-smooth error estimate of the initial data reflects the singularity at t = 0 (see Theorem 4.3. 2), we were particularly interested in the behavior of the errors when T is close to zero.
Concluding Remarks
We consider two completely separate schemes based on the convolution quadrature, namely backward Euler (BE) and second-order backward difference (SBD). The main advantage of using the convolution quadrature is that it allows the discretization of the derivative and the integral term in (5.1) simultaneously. The semidiscrete scheme is based on the piecewise linear GFEM in space, and the fully discrete scheme is based on the convolution quadrature in time with the generating functions given by the BE and SBD methods.
Analysis for the BE Method
This is achieved by reformulating the problem by transforming the Caputo fractional derivative into the Riemann-Liouville, and then applying convolution squaring. As usual, the smooth data error analysis is performed for the non-homogeneous problem, while the homogeneous problem is considered for the non-smooth data error estimation. We are now able to provide the error estimate of the BE method for both smooth and non-smooth initial data.
Analysis for the SBD Method
Analogous to Lemma 5.2.1 for the BE method, we have the following lemma regarding the SBD scheme for the convolution quadrature (cf. [39, Theorem 5.2, p.144]). The error estimates for the fully discrete SBD method for both smooth and non-smooth data are presented in the following statement. For smooth initial datau0, a bound of the first termu(tn)−uh(tn) is given by Theorem 3.3.2.
Numerical Illustrations
For the BE method, Tables 5.1 and 5.2 respectively show the convergence rates (CR) for smooth data (case (a)) and non-smooth data (case (b)), respectively. Our numerical experiments reveal O(k) accuracy in time, as expected from the theoretical results of Theorem 5.2.1 (see Tables 5.1 and 5.2).
Concluding Remarks
We establish the results of the Sobolev regularity of the solution for the inhomogeneous problem with respect to smooth and non-smooth initial data u0, including u0 ∈ L2 (Ω). Furthermore, the L1 scheme for temporal discretization is analyzed and the O(k) order error estimates for homogeneous and inhomogeneous problems are proved. Using piecewise linear functions in space and BE and SBD schemes in time, the authors have derived optimal order error bounds for semi-discrete and fully discrete schemes for homogeneous and inhomogeneous problems for non-smooth problem data.
Regularity Results of the Solution
Using the Laplace transform, the authors first plotted the solution along a suitable contour in the complex plane and established some stability and regularity results of the solution. The same authors extended the convergence theory of the SE to space-time fractional diffusion problems. Weakly singular kernel 94 To continue this work, it is important to obtain a limit of the function l.
Error Analysis for the Semidiscrete GFEM
Using the above limits, we proceed as previous calculations to conclude kI1k ≤C. Heru0h is an approximation of u0 inSh and the operator Ah :Sh →Sh is the discrete version of the operator A defined by. We now state the semidiscrete error estimate of the non-homogeneous problem with zero initial data.
Error Analysis for the L1 Scheme
Auxiliary Results
Let us recall the following estimates from [26, Lemma 3.1] of the function g(z) := (1−e−zk)/k on the contour Γk; there are two positive constants C1 and C2 such that. The Weak Singular Kernel 105 It is easy to see that the limits of the function |g(z)| in (6.31) remain true if we extend the contour Γk to some extent to the right in the complex plane. For later use in our error analysis, we now compute the error between the functions (1−ekα−zk)ϕ(zk) and del(z).
Error Estimates for the Homogeneous Problem
Weak singular kernel 111 Since vh(tn) −Whn = uh(tn)− Uhn, the proof follows by the triangle inequality and appeals to the limits of I5 and I6. This is in contrast to the result obtained in [71, Theorem 3.1], where the author has established a suboptimal error estimate with a general second-order memory kernel and a certain β, namely β = 1/2.
Error Estimates for the Nonhomogeneous Problem
The Weak Singular Kernel 114 The following lemma derives the error estimate when the source term fh is of the form fh = 1∗fh0 with fh(0) = 0. Now Lemmas 6.4.7 and 6.4.8 present the error estimate for the time discretization for the inhomogeneous problem with zero initial data. The Weak Singular Kernel 117 We are now ready to give fully discrete error estimates for scheme (6.24) for homogeneous and inhomogeneous problems.
Numerical Illustrations
To calculate the rates of convergence (CR) of the discrete solution, we use the L1 scheme for the discretization of the fractional derivative and the quadrature convolution scheme BE for the Volterra integral term. We first report the CR for the spatial discretization and then the same for the temporal discretization for cases (a), (b), and (c). The empirical CRs are listed in Table 6.3 and are compatible with the theoretical results of Theorem 6.3.2.
Concluding Remarks
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