Convergence in 𝘩

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To observe convergence in \(h\), we calibrate \(\tau\) and \(m\) to \(h\) as \(\tau \sim h^\alpha\) and \(m \sim \lceil\log_e^2(h)/(\pi^2(1-\alpha/2))\rceil\). This ensures that the total convergence rate with respect to the mesh size \(h\) is of order \(\alpha\), i.e., \(\|u-U_{h,m}^\tau\|_{L_2((0,T);L_2(\Gamma))} \leq C_hh^\alpha\). This can be verified by estimating the slope \(S_h\) in the regression \(\log_{10} E = S_h\log_{10} h+\log_{10} C_h\), where we expect \(S_h\sim\alpha\).

# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(rSPDE)
library(MetricGraph)
library(grateful)

library(ggplot2)
library(reshape2)
library(plotly)

capture.output(
  knitr::purl(here::here("functionality.Rmd"), output = here::here("functionality.R")),
  file = here::here("purl_log.txt")
)
source(here::here("functionality.R"))

# Parameters
T_final <- 2
kappa <- 4
N_finite = 4 # choose even
adjusted_N_finite <- N_finite + N_finite/2 + 1
# Coefficients for u_0 and f
coeff_U_0 <- 50*(1:adjusted_N_finite)^-1
coeff_U_0[-5] <- 0
coeff_FF <- rep(0, adjusted_N_finite)
coeff_FF[7] <- 10

AAA = 1
OMEGA = pi

# Time step and mesh size
POWERS <- seq(14, 9, by = -1)

time_steps <- 0.1 * 2^-POWERS
h_vector <- time_steps^(1/2)

# Overkill parameters
overkill_time_step <- 0.1 * 2^-14
overkill_h <- (0.1 * 2^-14)^(1/2)

# Finest time and space mesh
overkill_time_seq <- seq(0, T_final, length.out = ((T_final - 0) / overkill_time_step + 1))
overkill_graph <- gets.graph.tadpole(h = overkill_h)

# Compute the weights on the finest mesh
overkill_graph$compute_fem() # This is needed to compute the weights
overkill_weights <- overkill_graph$mesh$weights

m_values <- c()
alpha_vector <- seq(1, 1.8, by = 0.2)
# Create a matrix to store the errors

errors_projected <- matrix(NA, nrow = length(h_vector), ncol = length(alpha_vector))
for (j in 1:length(alpha_vector)) {
  alpha <- alpha_vector[j] # from 0.5 to 2
  beta <- alpha / 2

  # Compute the eigenvalues and eigenfunctions on the finest mesh
  overkill_eigen_params <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = overkill_graph)
  EIGENVAL_ALPHA <- overkill_eigen_params$EIGENVAL_ALPHA # Eigenvalues (they are independent of the meshes)
  overkill_EIGENFUN <- overkill_eigen_params$EIGENFUN # Eigenfunctions on the finest mesh
  
  # Compute the true solution on the finest mesh
  overkill_U_true <- overkill_EIGENFUN %*% 
    outer(1:length(coeff_U_0), 
          1:length(overkill_time_seq), 
          function(i, j) (coeff_U_0[i] + coeff_FF[i] * G_sin(t = overkill_time_seq[j], A = AAA, lambda_j_alpha_half = EIGENVAL_ALPHA[i], omega = OMEGA)) * exp(-EIGENVAL_ALPHA[i] * overkill_time_seq[j]))
  
  for (i in 1:length(h_vector)) {
    h <- h_vector[i]
    time_step <- h^alpha
    m <- min(4, ceiling((log(h))^2 / (pi^2 * (1 - alpha / 2)))) # min(4, ceiling(25 * (log(h))^2 / (16 * pi^2 * (1 - alpha / 2))))
    m_values <- c(m_values, m)
    graph <- gets.graph.tadpole(h = h)
    graph$compute_fem()
    G <- graph$mesh$G
    C <- graph$mesh$C
    L <- kappa^2*C + G
    eigen_params <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = graph)
    EIGENFUN <- eigen_params$EIGENFUN
    U_0 <- EIGENFUN %*% coeff_U_0 # Compute U_0 on the current mesh
    Psi <- graph$fem_basis(overkill_graph$get_mesh_locations())
  
    time_seq <- seq(0, T_final, length.out = ((T_final - 0) / time_step + 1))
    my_op_frac <- my.fractional.operators.frac(L, beta, C, scale.factor = kappa^2, m = m, time_step)
    INT_BASIS_EIGEN <- t(overkill_EIGENFUN) %*% overkill_graph$mesh$C %*% Psi
    # Compute matrix F with columns F^k
    FF_approx <- t(INT_BASIS_EIGEN) %*% 
      outer(1:length(coeff_FF), 
            1:length(time_seq), 
        function(i, j) coeff_FF[i] * g_sin(r = time_seq[j], A = AAA, omega = OMEGA))
    
    U_approx <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
    U_approx[, 1] <- U_0
    for (k in 1:(length(time_seq) - 1)) {
      U_approx[, k + 1] <- as.matrix(my.solver.frac(my_op_frac, my_op_frac$C %*% U_approx[, k] + time_step * FF_approx[, k + 1]))
    }
    
    projected_U_approx <- Psi %*% U_approx
    projected_U_piecewise <- construct_piecewise_projection(projected_U_approx, time_seq, overkill_time_seq)
    errors_projected[i,j] <- sqrt(as.double(t(overkill_weights) %*% (overkill_U_true - projected_U_piecewise)^2 %*% rep(overkill_time_step, length(overkill_time_seq))))
    print(paste("m =", m, ", alpha =", alpha, ", h =", h, ", time_step =", time_step))
  }
}
save(errors_projected, file = here::here("data_files/errors_projected_h.RData"))

# Load the errors_projected data
load(here::here("data_files/errors_projected_h.RData"))

observed_rates <- numeric(length(alpha_vector))
for (u in 1:length(alpha_vector)) {
  observed_rates[u] <- coef(lm(log10(errors_projected[, u]) ~ log10(h_vector)))[2]
}

theoretical_rates <- alpha_vector


p_h <- error.convergence.plotter(x_axis_vector = h_vector, 
                                 alpha_vector, 
                                 errors_projected, 
                                 theoretical_rates, 
                                 observed_rates,
                                 line_equation_fun = loglog_line_equation,
                                 fig_title = expression("Convergence in " * italic(h)),
                                 x_axis_label = expression(italic(h)))
p_h

Figure 1: Comparison of theoretical and observed convergence behavior for the \(L_2((0,T);L_2(\Gamma))\)-error with respect to \(h\) on a \(\text{log}_{10}\)–\(\text{log}_{10}\) scale. Dashed lines indicate the theoretical rates, and solid lines represent the observed error curves. The legend below each plot shows the value of \(\alpha\) along with the corresponding theoretical (‘theo’), and observed (‘obs’) rates for each case.

ggsave(here::here("data_files/conv_rates_h.png"), width = 4, height = 5, plot = p_h, dpi = 300)
save(p_h, file = here::here("data_files/p_h.RData"))

## [1] "Successfully connected to Slack"

1 References

cite_packages(output = "paragraph", out.dir = ".")

We used R version 4.5.0 (R Core Team 2025) and the following R packages: gsignal v. 0.3.7 (Van Boxtel, G.J.M., et al. 2021), here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), knitr v. 1.50 (Xie 2014, 2015, 2025), Matrix v. 1.7.3 (Bates, Maechler, and Jagan 2025), MetricGraph v. 1.5.0.9000 (Bolin, Simas, and Wallin 2023a, 2023b, 2024, 2025; Bolin et al. 2024), patchwork v. 1.3.1 (Pedersen 2025), plotly v. 4.10.4 (Sievert 2020), RColorBrewer v. 1.1.3 (Neuwirth 2022), renv v. 1.0.7 (Ushey and Wickham 2024), reshape2 v. 1.4.4 (Wickham 2007), rmarkdown v. 2.29 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2024), rSPDE v. 2.5.1.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), scales v. 1.4.0 (Wickham, Pedersen, and Seidel 2025), slackr v. 3.4.0 (Kaye et al. 2025), tidyverse v. 2.0.0 (Wickham et al. 2019), viridisLite v. 0.4.2 (Garnier et al. 2023), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

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