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# Questions tagged [convergence]

Questions related to whether the sequence of iterates generated by an iterative method has one or more limit points, and if those limit points have the correct properties.

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### Confused on how Method of Manufactured solutions works?

I am new to computational science and I am trying to wrap my head around how MMS works. I am solving the time independent Helmholtz equation as a simple test of the technique so my starting equation ...
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### Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous

I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too. ...
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### Step3 in deal.II - Convergence of the mean

I'm trying to understand the Convergence of the mean part of the Step-3 tutorial in deal.II. The authors say that $\frac{1}{|\Omega|}\int_{\Omega} u_h(x)dx$ converges with $\mathcal{O}(h^2)$, but I ...
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### what is non-asymptotic convergence?

I guess convergence in general means it is in asymptotic sense but what does non-asymptotic convergence mean?. Can someone please explain with an example?
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### Convergent Finite Difference Scheme for Parabolic Equation

Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$ where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$. In Strikwerda's book, the ADI scehme \begin{align*} \left( 1 - \frac{...
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### Non-conforming bi-linear finite element

The four-noded bi-linear rectangle element, which sometimes goes under the name Melosh element, is non-conforming unless the element sides are aligned. Out of curiosity I have implemented this element ...
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### Best way to check if SOR solution has converged for 2d matrix

I have written a SOR algorithm to solve the Laplace equation on a 2d grid. The outside of the grid is fixed at 0 and the central square is fixed at 10. I can obtain the fully converged solution for ...
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### Determine stability of an algorithm?

This is related to a question I answered on Stack Overflow regarding calculating the square root of a number. I was thinking about it and realized that the formula is just the first in a family of ...
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### Testing the time dependent Schrodinger Equation with an analytical solution?

I am numerically solving the Schrodinger Equation in 1D first and in higher dimension later, but I want to know the convergence rate of my numerical solver in different grid size and numerical methods....
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### Difference between contraction rate and convergence rate

I am slightly confused about the concept of contraction rate. For me it sounds equivalent to a convergence rate. Could somebody clear up the difference for me? I have those two definitions of ...
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### Abaqus, ANSYS, and FVM solver for thermal expansion problem converges to different values

Is it reasonable for a FEM and FVM code to converge to slightly different solutions for the same physical problem (identical BCs, geometry, properties, etc...), provided stability constraints are ...
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### Blowup of error in Conjugate Gradient method with periodic Dirichlet Poisson matrix

My problem is that the L2-Norm of the residual for the periodic Poisson matrix $P$ is initially decreasing but starts to blow up after a certain number of iterations. The blowup happens earlier the ...
I'm working with a numerical algorithm whose output $y$ asymptotically approaches a certain unknown value $a$. I expect an exponential convergence, i.e. the data $y$ given by my algorithm should be ...